No there were 12 like the Enterprise (Constitution class) at the time of the episode "Tommorrow is Yesterday" when <PERSON> utters the line: "There are only twelve like it in the fleet."
The Star Trek Writer's guide established there were fourteen Constitution Class starships: Constitution, Constellation, Enterprise, Yorktown, Lexington, Farragut, Republic, Intrepid, Exeter, Hood, Excalibur, Valiant, Kongo, Potemkin. The original AMT model decal sheet validates these names.
The Constitution Class had been around for at least 13 years and probably longer (The Menagerie) and <PERSON> had served aboard at least two of them before assuming command of the Enterprise. The USS Republic (Court Martial) and the USS Farragut (Obsession.)
It is almost certain the Farragut was lost or destroyed when it encountered the Cloud Creature, and that would make <PERSON>'s statement in Tomorrow is Yesterday true when he said it. There were only twelve like it (that were left) in the fleet.
The Star Trek Reference manual lists four as destroyed. We saw at least two of them destroyed, Intrepid in Immunity Syndrome and Constellation in Doomsday Machine. These episodes occurred after Tomorrow Is Yesterday. We heard of the destruction of Farragut in Obsession. 126
We also probably saw the destruction of the Excalibur in Ultimate Computer but it was never established that it was destroyed beyond salvage.
This leaves the Valiant. This is problematic. It may be bad writing. Other than the ship named 'Valiant' that disappeared two hundred years before the episode 'Where No Man Has Gone Before' we saw no Constitution class USS Valiant. We did however see a USS Defiant (Tholian Web) that wasn't on the writer's guide list. Like the other ship names used from history there was an actual British battleship named Valiant. There was never an actual ship named 'Defiant.' There was a fictional HMS Defiant in the film 'Damn The Defiant' I think somebody messed up, meant to type Valiant but chose to type Defiant instead and it became hard to explain canon.
Either way there were fourteen original Constitution Class and thirteen left when <PERSON> made his statement in Tomorrow Is Yesterday.
And they were all Constitution class. The model gave you a Constitution class starship and 14 names to choose from. 126