If an Andorian military academy would mean that SF is a military organization, then the Vulcan science academy would contradict that and mean SF is a scientific organization. One planet's academy doesn't define the whole fleet's role.It's also Andorian, which we know are a very martial people.
Memory-Alpha says it trains people for service in the Andorian military but also offers courses to Starfleet Officers. But that seems like speculation, I don't see anything on the dipolma that says that, unless it's part of the unreadable text.
It does say it's part of the Department of Foreign Studies, which seems to imply it isn't a Starfleet institution. but who knows. It's just a prop.