Basically what the title says. I've only played two games like this so far (Silent Hill 2 and OMORI) and am looking for more.
Could be games, ROM hacks, mods of existing games, etc.
EDIT: So many great suggestions! Getting a bit overwhelmed by how many there are, so I'm going to need to find a way to put this in some kind of list with an order to play the games in. By all means, keep 'em coming, though!
Brothers: a tale of two sons. You play as both brothers, working together to find a cure for their father sickness. Pretty sure I traumatized my SO with this one...
Came here for this - thisgames was amazing and I totally did not see the ending coming until just before it happened. Heart wrenching and bittersweet - such a mind-blowing game!