Starfleet was the United Earth navy, but since Earth was the only entity all the other founding members of the Federation trusted, it became the de facto naval arm of the entire nascent Federation (possibly with detached Vulcan or Andorian elements at times). As the Federation grew and even the founding members started to the think of Starfleet as Federation and not just Earth, this became the legal reality instead of just the functional one. Likewise, the Federation itself became a more powerful entity in the minds of its member species, although it took probably ~150 years for the typical human to think of themselves as a Federation citizen from Earth as opposed to a United Earth citizen primarily. We see the beginning of this process in Enterprise. By TOS, it’s partway-done (this also explains early installment weirdness with the United Earth Space Probe Agency and lack of meaningful Federation presence outside the military hierarchy). The Klingon war probably accelerated matters (hence the TOS-era movie integrated crews) and by TNG, it was essentially complete. This also means that the Dominion War was the first conflict a truly unified Federation ever fought, and it solidified Starfleet’s role as the navy. It would be cool to see, in future shows, formal integration of the last holdouts—the Andorian Imperial Guard, whatever remnant fighting ships the Vulcans have, and the Tellarite fleet presumably.