It's less interesting than you might think. Universities are essentially medieval institutions. The academic dress (or regalia) is a fancied-up version of the common clerical wear people wore back in the day. Much like a ball gown is fancied up sundress or whatever. Note that the only other medieval institution still around is the catholic church and it's offspring. They wear very similar garb for ceremonies.

Well not really. Cambridge started as a dancing school for instance. You'd send your boy off at 11 or 12 and he'd come back a man at 14 ready to pitch woo at the ladies having mastered the courtly dances. It was the dress of non-peasants at the time and universities and the church had that in common. 1000 years from now they might be wearing three piece suits for the same reason.