There are a lot of magic school series, but that kind of goes against the "discover" requirement, since it's already well known enough to have a school for it.
The Fate series has women who use magic, though men use it too. Start with Fate Stay/Night.
Gonna show my age, but Chobits fits this bill very well, though the magic part is a very slow burn (and is less "learned" than... uncovered). It's also debated whether it is actually magic or just technology, but imo it's very clearly magic.
Sousou no Frieren has multiple arcs based around (mostly women) mages learning magic. It's also one of the best mangas or animes in decades, imo.
Mai HiME is a classic magical girls series that has great characters and is literally all about them "discovering" their individual powers. I actually recommend the anime over the manga, which shifts the POV from a male protag to Mai, his love interest, who is the real, titular MC.
I showed it to my wife and she loved it. My Otome, not so much (she kept asking when Mai was going to show up again, and was very sad when it was just the very end).
Chobits mentioned... cute story, but all the explanations are technology based, no magic needed (except "the magic of love", hm). It's more of an exploration of our upcoming IRL approach to humanoid robots with an AI, and what will it take to train them for human relationships: a humanoid training platform learning from human interactions.
Mai HiME is a classic magical girls series that has great characters and is literally all about them "discovering" their individual powers. I actually recommend the anime over the manga, which shifts the POV from a male protag to Mai, his love interest, who is the real, titular MC.
Is the magic innate or can anyone learn it because I prefer magic systems where anyone can learn it
Good distinction. In my younger days when Anime choices were slimmer, I enjoyed Sailor Moon, and Nanoha. Nowadays, I can't remember the last time I watched a transforming Magical Girl anime, I much prefer the later. Fantasy settings, magic, Isekai (...which are abundant nowadays....bordering on too abundant, but I don't get tired of them because magic).
Edit: BTW, not a knock on Nahoha. As I recall, great series, though only gave it a once through. ... Now that I'm thinking about Magical Girl anime, also remember Figure 17 I think it was called... still have music from it's OST on my playlist; had some good battle music, and calming/relaxing music.
not exactly "magical girl" genre, Little Witch Academia is pretty good. More akin to Worst Witch? Its aimed at younger readers of course but its still fun.
I have already watched Little Witch Academia and enjoyed it so much its one of my top anime and I haven't found anything that can scratch the same itch as it did
I’m looking for magical girl manga where the girls discover and learn magic by themselves instead of it being granted through some contract
Witch Hat Atelier has girls learning magic by being taught it. It's not really the magical girl genre -- and their mentor is a guy and there are plenty of male witches in the background -- but the overall focus is on a small group of young girls learning magic.
Not sure if that's really what you're looking for, but its at least adjacent! (Also, the art is lovely.)
e: Oh, and there's an upcoming anime of it, here's the trailer
I kinda loved Miracle Girls when I was a teenager. In retrospect, I think it was mostly because identifying with the characters gave me gender euphoria. I don't remember a ton about the details, but it was fun and sort of slice-of-life wholesome. That was in the early 00s so there may be some shockingly outdated stuff in there that I just glossed over and don't remember, but I feel like that's probably just my reluctance to recommend anything 30 years old uncritically!