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.