As Enterprise puts it, Starfleet was once shorthand for the United Earth Space Probe Agency Starfleet (of Star Fleet)...
Star Trek has always been a bit unclear as to whether (even in post-Federation/post-Enterprise times) Starfleet is a Federation "navy" or an Earth navy.
No, it really hasn't. The only times Starfleet has ever been referred to as an Earth "navy" were in early TOS, before they created Starfleet for "Court-Martial" and the Federation for "A Taste of Armageddon." Since then, Starfleet has been explicitly stated on multiple occasions to be a Federation
agency, answerable to the Federation
In Star Trek 6 and Into Darkness, we saw a roomful of Starfleet admirals - and they were all human.
In STID, we saw several non-Human admirals in that briefing room scene, actually. And of course, we never know:
1. How many of those Human-looking people actually are
Human, or how many of them are non-Humans whose species looks
2. How many of those Humans are actually from Earth
as compared to, say, former Earth colonies have have become their own separate, independent Federation Members, or to other planets that host large Human populations. For all we know, three or four Human extras in any given scene may be from Andor or Tellar.
Meanwhile, a preponderance of one species at one location does not make that organization an agency of that species's homeworld's government. If Starfleet's Tellar headquarters and Tellar-based fleet is mostly staffed by Tellarites, that doesn't mean Starfleet is a Tellarite agency.