There was a Starfleet in the Earth-Romulan Wars. This was established in the episode "Balance of Terror".
There was a Captain Stiles in the space service then.
The term "space service" could mean anything. It doesn't imply a specific name being used. It's totally generic. Remember, 'Starfleet' hadn't yet been chosen as a name when that episode aired (that's why the early TOS eps had a wide assortment of often comical pulp-ish names like "Space Command" and "Star Service" and all that).
In any case, it seems obvious that the two Starfleets must be different, for at least two reasons:
- Military organization. The original Earth Starfleet isn't military, but the Federation Starfleet is.
- Jurisdiction. A Starfleet that is reponsible for Earth alone cannot possibly be the same one that holds sway over hundreds of Federation member worlds. We have no idea why the Federation chose the name 'Starfleet' for its military, they must have thought it sounded good or something.
As for why (out of universe) the writers of ENT chose the name Starfleet? I don't know the answer to that either. They may have been ordered to do so by the network (same reason why transporters exist in that show - they didn't want to have them, the network made them do it).