captain among many in Starfleet,
Each of those "many" would be special people, especially if we are only counting captains of what may be lead ships in groups spread about a week's travel apart see what I wrote below).
Aren't McCoy's ex-wife-the mother of his daughter, and Nancy Crater the same person? The show does not say they were married, but when TAS said he had a daughter, it would make sense, that assuming he had been married, Nancy Crater would be the person he'd have been married to.
Ron Tracey or Matt Decker could be in same quadrant or region.
In other places I have suggested that perhaps there are small fleets of around four ships that operate in a Quadrant (this being a 4th of a sector not a 4th of the whole Galaxy), and that perhaps each fleet uses ships with 3-4 different roles such as depicted in the FJSTM. Thus Kirk might know John Daly of the Astral Queen because the Astral Queen is the transport ship in the small fleet of which the Enterprise, being a cruiser, is the lead ship.
As such, Matt Decker, Ron Tracy, and Bob Wesley, and the captain of the Defiant, are a kind of "neighbors," being the commanders of lead ships in the nearby quadrants. Kirk might at least vaguely know around 16 captains--commanders of the cruiser, and/or cargo carrier, tanker, surveyor and scout in each of the nearby quadrants.
If there was a war, these would be the people he'd be working closely with the create a defense perimiter or intimidate the enemy.
In the case of the Defiant, if its number of 1764 is to be accepted, it might even be that that ship is newer but not meant for front line duty, is not a cruiser and looks externally like the Enterprise, despite serving in a different way.