According to "Bread and Circuses", everything else was just a spaceship and far inferior in every way.
Compare to modern Star Trek, where Enterprise NX-01 a century before and the USS Discovery in the same era can do everything the classic Enterprise could (and much more, in the latter's case)
Your memory is failing you, or else you are confusing three separate things and assuming that they are the same thing, even though there is no proof that the three things are identical.
"Tomorrow is Yesterday":
KIRK: Bridge.
CHRISTOPHER: Must have taken quite a lot to build a ship like this.
KIRK: There are only twelve like it in the fleet.
Does Kirk's 12 include or excluded the
Enterprise? Does Kirk mean there are only 12 ships vaguely resempling the
Enterprise, or that there are only 12 exactly like the
Enterprise? Which fleet are there only 12 like the
Enterprise in? The entire Federation Starfleet, the United Earth Space Probe Agency fleet, the fleet patrolling sector K, or some other fleet or sub fleet?
"Bread and Circuses":
CLAUDIUS: You're a clever liar, Captain Kirk. Merikus was a spaceship captain. I've observed him thoroughly. Your species has no such strength.
MERIK: He commands not just a spaceship, Proconsul, but a starship. A very special vessel and crew. I tried for such a command.
There is no direct statement whether the 12 like the
Enterprise are identical with the starships that are so much superior and special according to Merik. Perhaps the 12 like the
Enterprise are a tiny subset of Merik's starships.
And what about Constitution class starships? There are no mention of starship classes in TOS. The plaque on the bridge describes the
Enterprise as a Starship class vessel - if anyone can read it onscreen to make it canon. In "Space Seed" Khan asks to study technical manuals of the
Enterprise, and later is seen viewing a schematic of a phaser for a Constitution class ship - if anyone can read it onscreen to make it canon.
As far as I am concerned there is no need for anyone to believe that the 12 like the
Enterprise, starships, and
constitution class ships are the same. They might be two or three different and perhaps overlapping or nested categories.
So nobody knows how many starships there were in the era of TOS.