Hogwarts and D&D are 2 big influences in manga/anime. Lots of properties have characters who specialize in one type of magic or another. Often the magical system is kind of vauge. The following is non-definitive, corrections welcome.
Full Metal Alchemist - Magic is treated a lot like technology, following from first principles. Anyone can do it I guess, though alchemy is a specialty practiced by few.
Frieren - some is intuitive, some follows logic. I don’t know of any limitations regarding who can cast it.
Hunter x Hunter: You are taught how to use Nen if you pass a ridiculously difficult and competitve test.
Fairy Tail - not sure, I think it’s inborn though. It’s not clearly stated, but over at Reddit one poster opined: “People learn it, this is shown throughout the show, tho some people will have more of a natural affinity for it.” Might this be a default?
One Piece - eat a devil fruit, get a power and a downside.
JoJo’s Bizarre Adventure - meet a stand (supernatural being) and he becomes your familiar, gain powers.
Naruto - some have more potential than others. I have the sense that some magic (ninjitsu) is open to most, though 2 characters have zero magical potential.
The Last Avatar - can all members of the water-bending tribe bend water? Ditto for other 3 elements.
Ok, I’m a Spider, So What? I think you need a skill and mana to put into it. The world is gamified: you gain skills by doing stuff. I’m not sure how much magic a peasant would pick up.
Flying Witch, Ancient Magus’ Bride - definitely born into it.
Death Note - find a death notebook, win a death god who follows you around; keep the magic notebook.
A Certain Magical Index. From fandom: “In contrast to espers who have only one ability that is decided by birth, magicians can create many supernatural phenomena without restrictions, but they have to take the time and effort. Additionally, all forms of magic are based around a certain ideological premise. This kind of premise can originate from almost anything culturally narrative including religions, belief systems, mythologies, or even folklore and fairy tales”. I think it’s potentially open to all, though, “A normal person will suffer more severe consequences than a magician if they fail,” and some lack magical power. IOW they fudge the issue, not unusual.