Here's what happened:
Before ENT ever even aired, a summary of the first episode's script was posted online, mentioning among other points that the heroes were Starfleet and they'd make first contact with the Klingons. At the time, many people were using the Okudas' Chronology
for reference, and it said that Starfleet was established in 2161 with the UFP and that first contact with the Klingons took place only 50 years before TOS. Cue a bunch of people crying "continuity error" and then trying to come up with explanations for these "inconsistencies."
But they never were inconsistencies to begin with, since the timeline of these events had never actually been pinned down in canon and the authors of the Chronology
had merely conjectured based on the little specific information available. People simply hadn't yet adjusted their thinking to accommodate the new data presented in the new series, leading to this fandom notion (never espoused on the show or by anyone directly involved with it) of two Starfleets.
Don't believe me that this is how the whole idea came about? The record is out there on the internet, probably on this very board and certainly on others
. It's pure "fanon."
It was natural to presume that Starfleet was founded at the same time as the Federation before we were shown otherwise because up until then we'd always seen SF as part of the UFP, and the Starfleet Academy logo originally seen in "The First Duty" (TNG) had MMCLXI (2161) on it. But in retrospect, that really need only be seen as indicating that the San Francisco campus of Starfleet Academy
was founded in 2161.