It's hard to say if there's any such institution in Eorzea. From what I gathered when playing, I think the safest way to approach education is to imply the guild's take care of that. Upbringing is most likely handled by the family itself until a certain age, at which point I'd expect the child to either become an apprentice on one of the guilds or to start learning their parent's trade if they can't afford the apprenticeship for whatever reason. Guilds don't seem to ask for money to the player-characters (maybe they have a special 'adventurer scholarship'?), but the teachers would still have to be convinced that the potential apprentice has...well, potential.
Alternatively, we can assume there are off-screen schools and academies, but then one would expect them to come up during the game: all the scholarly NPCs I have seen always come from one of the guilds.
Alternatively, we can assume there are off-screen schools and academies, but then one would expect them to come up during the game: all the scholarly NPCs I have seen always come from one of the guilds.