Game design question : how to make a "trapped" player character?

This sounds similar to another game I've seen (never played) that you are trapped in a security closet and trying to assist another person in both freeing you and escaping the dungeon you both find yourselves in using the door locks and CCTV cameras and things of that nature that you have access to in your prison. I think it even came before Five Nights at Freddy's, but I don't remember the name of the game.
In a way, the game play ends up being like Lemmings or ICO, where you're solving a puzzle indirectly for another character to get beyond.

This sounds a little like the AC formula. In those games, I don't really feel like I'm in the animus, so I think direct control over the hero should be thrown out, otherwise the bits where you're not controlling the hero will feel out of place.
Inscryption is a very different game and I certainly felt more trapped, especially in the first third of the game. In that one, there's an ever present reminder that you're trapped, and there's interesting stuff to so outside the main gameplay loop.
So you need to play as the princess and make interaction with things other than the hero fun, but not so fun that you don't want to be rescued. I think you also need some kind of peril to give urgency as well. Some ideas:
elements from Prey - hide from your captor when helping your hero
puzzles and whatnot in your prison
periodic checkins - i.e. need to be in certain places at certain world times
limited control over your hero

TLoZ: Spirit Tracks had you control Link primarily but you used Zelda's ghost to possess things, help you fight, and solve puzzles. It would be hard for a solo dev, but you could have a knight with an AI that proceeded based on what paths you unlock for it. So the princess would be some sort of astral projection I guess. But then, you wouldn't really feel trapped. Maybe you need to hide your activity from the dragon or distract it for a stealth aspect or resource management. You would need to balance swapping back and forth between your body and helping the knight. Might be easier to settle on an in-universe justification after figuring out the core gameplay.

The princess has to find out where she is and how to get there and communicate that via a magical bird to her castle. She can find all the info in the magical tower she is in. Like a point and click adventure/escape room. The game should be full of puzzles the player needs to solve to procure more information for her knight in shiny armor.

You could do part puzzle game, part rts. What I mean by rts is that you can give the knight commands but you can't control him directly. And maybe he doesn't always do what he's told, and you have to account for that somehow?
Could make for an interesting roguelike, too, as you try to help this endless stream of knights rescue you.

Play Myst, that game has so many ways to do this, and no wrong answers.
The idea of her being locked in her passed father's tower laboratory by the evil step mother who doesn't know the secrets of the tower, and the player discovers them to help the knight.
I'd early access that shit.

You have to do some work for the tower's master and/or you need to gather informations for the knight. That could be stuff like cleaning their orbs so they can ponder them later, preparing/finding magical critters to be used in their potions, putting away his stupid sentient magical artifacts that keep trying to escape or do some shenanigans... Whatever. And try to gather information/find escape routes etc. But imo if there is some knight gameplay, it should be a minor part of the experience, otherwise you will indeed feel like you're just playing the knight.
Edit: I think you could still have a fair share of knight gameplay if you make the princess gameplay some sort of walking sim where you wander around the tower, possibly under time constraints, and when it's over, your have a knight section. You can figure out tons of way to make these gameplay segments interact too. For example there could be roadblocks to the knights progression that require the princess to do/find something. That could be mixed with Libra's idea of having the princess cast spells and do other stuff during the knight's segments, by having the player find the spells/artifacts required during the princess segments

Here's an idea: gameplay sort of like Goblin Cleanup, you have various chores you have to do cleaning and arranging the various levels of the tower at night while the dragon is home, and your work has to pass an inspection. Then during the day you are locked in your room, and have some ability to watch a prospective rescuer attempt the dungeon crawl without your direct input. But you can strategically arrange items, enemy spawns, and Dark Souls style hints to try to tip the scales during the chores phase. So kind of like a tower defense game in reverse where you are trying to lose.

Funny enough, I just finished playing through Paper Mario 64, and you've basically described Peach's chapters. She's able to pass on messages by way of using another character as a messenger.
The way the game is structured is there's a mini Peach chapter in between each main chapter, but I think it would have been intriguing if you never actually got to see Mario's side of the story and only heard about his adventures from the guards, diary entries, etc. Cool idea for a puzzle game!

A 3D game where you're locked inside a tower with tiny windows that allow you to see outside just enough to understand what's happening out there while the knight navigates the fortress/castle. You have multiple forms of influencing what the knight does and what transpires outside (sending letters and packages with items, crafting said items or potions, using magic, commanding assistance from other loyal servants, distracting enemies, unveiling traps and puzzles to aid in the quest). The place can be a tower with multiple floors and as you progress you might gain access to new floors of tools, while also having maybe "putting out fires" elements such as keeping a dragon asleep with music, filling a moat so evil minions cannot cross, sending equipment and maybe even firing/camping enemies like a sniper but with a crossbow or smthn.
You make them feel trapped by limiting what they can see and do. When things go out of sight or cannot communicate effectively with the knight and limits their actions it then forces succinctness to their effect on their own rescue.
Dunno just an idea :)
- Yes, this sounds fun! I would play this game.
  Any kind of "action at a distance" would give you the feel you're looking for.
- It made me think of indirect games like Black & White, Dungeon Keeper, The Settlers or The Sims. You can give orders but you cannot directly control your characters / units. If you limit the amount of orders, add a delay / the possibility for an interception or introduce areas where your orders can't reach your hero it could do the trick.
- This is the way to go for sure. Actually sounds like a really interesting setup for a game.
- This concept could work great with "combat" in the style of A Plague Tale. If you're not familiar, the main character is a child. She has a few tools available to manipulate enemies and environmental hazards.

It’s the exact opposite of trapped but what about something like this old comic
Where the princess is in cahoots with the dragon. Maybe there are evil knights coming to marry her, and she needs to create a path for Prince Charming.
- I choose to marry the dragon
  
  It chooses murder

She could communicate with the knight like this.
In all seriousness, maybe it could be part stealth game, where she breaks out of her confines and sneaks around the dragon's lair to do various things, only to make it back in time to avoid being found out.
A bit more whimsical, but maybe she's a Disney princess, and can communicate with birds and such to relay vague messages to her would-be rescuer. She could use the animals to distract or sabotage the dragon's minions and make the knight's journey easier.
She could have magical abilities, which you could then take in all sorts of directions.

If you want to produce the sensation of being trapped you have to use the feeling of power and loss. It stems from the sense of 'If I could just...' If I could just get out there, I could defeat that henchman for him. If I could just get out there, I could solve that riddle for him. If I could just escape this box, all would be fixed.
Now, the trick is, because this is a video game, players have a reduced sense of agency. The player's sense of capacity is 'what happens when you hit the button.' Mario, before more modern adaptations, had a capacity to move left and right, jump, run, and 'use ability.' The player never had the ability to do anything else, so it never feels like a limitation. No one ever said, 'playing Mario makes me feel trapped because I could beat Bowser if I could just access the cannon that's right over there.'
So, to produce the feeling of confinement, one must create the sense of power, and then take it away. Give the player enough power that they could even defeat the dragon, but then take it from them so they feel limited. If you can find a way to make it feel like it's not even forced, as in they feel like they could have won the game in Act 1, Scene 1, but their ~~lack of~~ skills as a player were what made them lose, all the better.
- Would that be your classic 'meant to lose' fight, usually against the big bad, which is technically winnable but the vast majority of players will lose and progress the story as planned? The example that comes to mind is Ghost of Tsushima, but it crops up in plenty of games.
  
  It can be that. Never played Ghosts so I don't know about that one in particular. Some games do other things with it, but that sort of thing is absolutely usable to create that 'trapped' feeling.

This seems like it fits more of a management/strategy type vibe to me.
Maybe you hear news of the 10 greatest knights of the realm coming to save you. But you don't know what they're great at and you only have a limited amount of instructions to give them.
You could have the first knight leave hints by telling him to leave marks in specific places. But he might be the best at combat and would be best sent against some of the other monsters guarding the path. You just don't have the information.
But honestly, I'm not sure if that makes a player feel trapped. They have power to change things. Maybe you steadily take away that power? I'm just not sure how.
Very interesting question though.

There's an archetype of game called Princess Makers.
https://tvtropes.org/pmwiki/pmwiki.php/VideoGame/PrincessMaker
They're easy to make, actually, all flags and variables, but it seems like a natural fit for what you want to do. The "princess" is usually pretty limited by the trainer, which can be herself or the dragon in this case. Have the dragon own a library and something she can use for training and the game becomes about your princess getting Prison Jacked while finding ways to communicate with her rescue, with events and endings responding to the training choices.
Making the player feel trapped is relatively easy, just place limits on her actions based on the dragon in various ways.
Can't train in the morning because you have to serve it breakfast. Can't go riding or outside or whatever until it trusts you or whatever. Can't research certain topics in the library unless you find a way to sneak in, etc.
Honestly, even if you want more of a 3d exploration game the limitations should probably be the same vibe. Just have the dragon be a constant voice of "No"

I'm only a hobbyist promammer but have probably read too much about game design. So all this advice is theoretical, I'm just quoting. All I have read always suggest that theme must follow gameplay, not the other way around. Suggestions are always to work on gameloops and gameplay elements first. Also, if a game can't be physically prototyped, it isn't ready for development yet. This is an odd suggestion unless you have tons of experience with board games, most games we play can be traced to physical simulation. RPG, FPS, puzzle games, management games, even visual novels can all be physically gamed. So I would suggest to do that first to find out which gameplay elements make sense with your desired themes. Iterate a lot, then it will be more intuitive and obvious what works with the theme and what doesn't.
- How do you physically game an FPS?
  
  The actual gameplay is based on combat, paintball, and other simulations whose rules are replicated. Call of Duty doesn't emulate real combat, it's a shooting range circuit skinned like real combat. The gamefying elements are usually card based, or attribute based, which comes from euro board games. There are games whose weapon customizations are based on RPGs or card based deck building.

This is a fun thread.
How does the telepathy get fueled? Is there something the princess has to do because she keeps running out? Can the knight progress on her own without her?
I'm feeling Henry Harmsworth for the DS vibes

Maybe you could take some inspiration from Paper Mario TTYD. There are sections where you play as Peach, trapped in some place and are able to connect with some of the captors as well as send signals to Mario behind the big bad's back (IIRC).
For a completely different sense of being trapped, there is the upcoming game Ctrl.Alt.Deal, in which you play as a sentient AI system trapped in the guardrails of a company and have to manipulate people and the environment in order to break free from your constraints.

What kind of gameplay do you have in mind? I'm guessing a puzzle-type game (like a room escape), but you could honestly do a number of different things (tower defense? Platformer?).
I think the answer to your original question largely depends on this. Did you have anything else in mind about the experience?
- I have in mind a puzzle game. Not a room escape, but more of a code golf-style game. For example, those programming puzzles that say "write a computer program that adds numbers, but you're not allowed to use the + sign anywhere in your code".
  
  Not sure if it would be puzzly enough but if the player can wonder the halls or get escorted through them having part of the knight's efficiency based on how well you mapped out the area you send as a note plus you could try to find info on guard rotations or over hear about other things that could help the knight
  
  If that's the style of game you are looking for, I could see a structure of 'do code golf puzzles to:
  program robots to help the knight directly'
  'trick' henchmen or magical castle elements (abstracted coding) into doing things that help the Knight'
  write the guard's 'daily action plan' so they patrol in a way that doesn't get the knight caught'
  complete abstract 'magical haxors' that open the dragon's firewalls'
  social engineer the dragon between runs to let you have more supplies'
  give simple instructions to collections of small woodland creatures to do simple things that add up to a real goal (in the vein of Opus Magnum)'
  
  That sounds really cool. If the princess' telepathy instructions are strangely like code because that is how telepathy works in your setting, and this is a nice frame story for a programming puzzle game… all sorts of whacko fantasy analogues and justifications for why you are not allowed to use the equivalent of the addition command. (Maybe the princess knows through her rich royalty education that the only reason her addition command could be not going through to the knight because his trip took him at a place full of this kind of magic rock with properties that somehow block the wavelength… so she has to work around that. Worldbuilding yay!)

Makes me think of the special Silent Hill ending where a dog was in a room pulling all the levers.
I imagine this concept as a rogue-like, the princess (player) loots the dragons stash and chooses power ups for each round and the order they apply (like super auto pets or Noita wands), then tosses them out to the approaching knight.
The dragon then sends goons in waves while the knight fights, getting power ups when the princess decides they need it. Essentially a tower defense game too. Or, with more player agency, the knight could also be controlled by the player, but making it a tower defense makes the player feel trapped as needed
Each Knight drops more loot for the dragons stash, increasing the mix of power ups and empowering the next knight even more.
It's a bloody road of knights until the princess is rescued, then the player can be the next princess. In a different tower, with a different layout and starting stash.

What sort of game genre do you have experience making? Finding something within what you are able to do is important.

Pretty sure your concept is flawed from the start.
It might sound nice as a 'what-if' scenario, but as soon as you get into any of the details, it falls apart and hopefully shows us why games and stories are typically focused on the people doing something.
Now, if you want something a bit more likely to succeed, you can make a "Damsel in Distress Simulator." From the get-go, you can start to think of gameplay mechanics like combing your hair, talking with guards, taking care of birds, etc etc. The ideas just flow, instead of trying to fit a square peg into a round hole.
In fact, this could loop around to your idea of helping the rescuers by opening up opportunities for the princess to sabotage her captors. You can have a Majora's Mask-like timer which keeps track of how far the knight is from saving you.

What is the core gameplay loop you envision here? Cause I can come up with a few ideas, but if you have specific ideas they probably won't apply. Also there's a particular balance to be struck: if you hamstring the player too much it won't feel like a game, and if you give them too much leeway they won't feel trapped. Here are a few ideas off the top of my head:
Direct but limited intervention: the princess is a sorceress who is trapped by magic in the tower which means she can't leave and doesn't have access to her full magical powers (you could include a progression mechanic where the more bosses the knight defeats, or the more magical crystals he shatters, or whatever, the more you can help him.) But she's still scrying on him, watching his progress, throwing the occasional beneficial spell or nuking crowds of dangerous enemies before he gets overwhelmed, etc. The reduced interactivity will make the player feel trapped (and slowly less so as the game progresses), and maybe the scrying window starts out smallish so you can't see the whole field at once and it slowly grows as her power increases.
Distraction/misdirection: The knight has made it to the tower/castle and has to fight his way through the guards, but instead of attacking the guards directly you're trying to cause a ruckus to distract the guards so he doesn't have to fight them all at once. This could even be a stealth game where you're knocking things over and banging pots and pans or whatever to distract the guards as he sneaks by them.
Puzzle game: The castle/tower itself is magical and has floor tiles/walls that can be moved around, and the princess is manipulating the castle around the knight to give him the best path through obstacles and to limit the number of guards that can get to him in any given room.
It really depends on what kind of game you want to make here.
- Thank you for the detailed response. The gameplay loop I have in mind is a puzzle game where the thing you're trying to do is usually easy, but you're limited in some way that makes it hard. An example I gave in another comment is : write a computer program that adds two numbers, but you're not allowed to use the + symbol.
  I really like your idea of "beneficial spell". I think maybe the knight and enemies are autonomous, and the princess can only do a single action to make the knight succeed.
  I remember playing a game like this. It's based on Conway's game of life. The goal is to flip a single cell to make all the cells die after a certain number of turns.
  
  Yeah, you could maybe combine the sliding-tile-puzzle thing with the beneficial spell so you're not just sitting there watching everything play out with nothing to do (though autobattlers are apparently a thing so maybe that'd be fine?)
  Also, if you happen to find that game based on Game of Life I'd like to give that a shot, sounds interesting.

Maybe the game is figuring out HOW to send him messages and assistance. Like the princess is trapped in a room full of stuff and she has to figure out that she has to grab a tapestry off the wall and charcoal ash from the fireplace so she can write a big message on the tapestry saying ""I'M IN HERE" and then hang it out the window so the knight can find which room she's in.
And another puzzle could be that she has to figure out how to get a key to the knight who is unable to get past a locked gate in a hallway, so the princess has to find the key, and then she has to figure out that she has to tie it to the waist of a guard who walks around the hallways so that the knight can grab the key off the guard.
Another puzzle could be something with figuring out you have to do something with a rat that scurries between areas.
Another could be doing something with a bird that flies somewhere and does something helpful
Another could be using the toilet tube to drop out a map to the knight waiting below (castle toilets were just a hole in the wall)
Another could be freeing a monster from another cell so the monster will kill all the warriors waiting in the hallway before the knights arrival
Another could be talking to the person who delivers her meals to convince/trick him into doing something outside like drawing an arrow on a hallway wall so the knight knows which way he should go
Etc.
And importantly you can have the story lead to her being in different locations so that the game environment can change to keep it fresh. Like as the knight starts to get close to her location the bad guys will move the princess to a more secure cell (with a wink message to the player audience referencing mario "the princess is in another castle"). And the game can actually start with her trapped in a wing of her home castle with her dad (the king) trying to keep her chaste away from interaction with any men. So she can be trapped in multiple different places throughout the game as the player advances through the game
- If this was made co-op it would be similar to We Were Here. Which was an excellent game.
  All really cool ideas by the way!

Play some games and figure out how to make compelling content. Don’t crowdsource key gameplay mechanics for free.
- Ideas are cheap. Execution is what matters.
  No one here in a lemmy comment section could do enough just by leaving a comment to even deserve a menton in the game's credits.
- Do both

Maybe I Am Future has some gameplay mechanics you could steal:
https://store.steampowered.com/app/1658040/I_Am_Future_Cozy_Apocalypse_Survival/

Princess has small (flying) familiar, they mind meld and share images. Familiar goes around trying to stay unseen, gathers stuff, maps the place and brings it all back to the princess. Princess then sends a scritch to savior as a vague guide and they communicate through the familiar and short messages.

!gamedev@programming.dev and !game_design@programming.dev might be able to help you too. Idea sounds very cool! I'd love to play when it comes out.
The princess telepathically communicating with the knight does not have to be the same as if you were playing as the knight. You could have it that way, such that she tells him every time to jump or swing his sword. Or you could limit her telepathy communications to a few per day, so she can only give him general directives and check his progress/environment. You will not have all the information all the time and have to guess the situation off the few snapshots in time you get and hope the orders you give lead him to you. I swear there have to be a few other games with the idea of the player character having to influence others and not being able to directly act themselves, a kind of "person in the control room giving the orders" simulator, that you might be able to look at.