Founded in 1088, the University of Bologna is probably the oldest university in the world; it is certainly the one from which we get the word 'university', although nobody knows the date the University of Oxford was founded, so that might be more ancient still.
Bologna University was apparently established by its students. In the city at that time there were groups of foreign students who were scholars of Roman law, and they banded together in mutual aid societies (usually by nation; this was apparently necessary because the city authorities exercised collective punishment on the basis of national origin) and hired people to teach them. Over time, these groups amalgamated into one "universitas".
Let's imagine a University of Magic. There is a city in which there is a relatively large number of magical practitioners. Students come from all over the world to study with them on an ad hoc basis, banding together in small groups and combining their funds to pay local mages to teach them. Over time, these different groups - and their mages - come to associate for mutual protection and profit, on a temporary basis at first, but eventually uniting into one organisation with a chancellor, different schools of magic (probably each founded by one of the different groups) and a huge library.
Hooks abound in such a setting for an "everything is mages" game. Obviously the PCs, as students, have field research to do. But there are also inter-school rivalries within the university itself, not to mention wider political conflict (the university of magic would be an institution of huge power, likely to come into conflict with other power bases in the local area). And, naturally, they need to find ways to fund their study...