It is pretty difficult to find nonhumanoids in Star Trek in general; I do not recall any instances of such in Starfleet uniform or Starfleet service, not even as dialogue mentions. That is, we never learned that the exotic but unseen Vilix'pran would have been nonhumanoid, say. But half the things we mistook for bushes at the San Francisco installation might have been cadets for all we know...
As for other installations, the Okudagrams do not mention competing, entirely separate Academies - and indeed one of the main purposes of the Academy might be to bring all the thousands if not millions of cadets physically together for patriotic propaganda. And never mind that this makes Riker's "eighth being best in class" almost astronomically significant, even if "class" means "Mrs Phillips' class" rather than "the class of 2347".
Other training facilities abound as Okudagram mentions, though. Mainly we hear of these in "Eye of the Beholder", the TNG rather than the TAS episode, that is. There's a Starfleet annex at Beta Aquilae II, Beta Ursae Minor II and Psi Epsilon III, a Starfleet OSHC, and a Starfleet Technical Services Academy. Distribution of this sort would make it statistically difficult for our heroes and random guest stars to refer to shared experiences at "The Academy" - so perhaps the off-Earth annexes concentrate on drastically different phenotypes, there mainly being molluskoids at Psi Epsilon III or whatever?
Timo Saloniemi