The only thing we have to go on to support this assertion is the visual evidence. Aside from the damaged Excalibur, all of the ships were obviously identical because they were filmed from the same model. The question is "Is being visually identical tantamount to being the same 'class'?"
Yes.
Part of the confusion is that the Enterprise is described in several ways: Starship Class, Constitution Class, and Mk IX Heavy Cruiser...and don't forget just plain "starship". My favorite way to interpret this so far is from the same issue of T Negative as Greg Jein's "Jonathon Doe Starship" piece. Ruther Berman posits that the Mk IX designation indicates the mission profile for which the ship was built. She goes on to list several other profiles (all made up, I think) with different Mk designations. The idea was that a ship might be outwardly identical to another, but it might have all sorts of different loadouts and machinery on the inside.
Think back to the period when TOS was in production. Space exploration was being performed in several series of craft; for the US, there was Mercury, Gemini, and then Apollo. Each craft in a series was more or less outward identical, but had some equipment differences. I've been thinking for a while that Roddenberry et al, being in the middle of that model of ship design, might have used in (consciously or unconsciously) as a model for Star Fleet design methodology, and that they thought that, if not all then at least a large number of Star Fleet ships would "look the same", but still have design differences internally. As support for this, Roddenberry justified his idea that all Star Fleet crew would be officers because they were the functional equivalent of astronauts, and at that time that was the model for who could go into space.
My opinion is that the Enterprise was a starship (capable of traveling between star systems), of the Starship class (fitted with the best of many things, if not everything, but especially including her warp drive), and a Constitution class (the current design of saucer, fitted with a secondary hull and two warp drive nacelles mounted to the secondary hull). In this world, we see many Constitution class ships in TOS, but we have no idea how many of them, if any, were Mk IX heavy cruisers fitted out for the long range exploration mission profile.
You're talking about open architecture. You can outfit different ships for different missions, but if you're putting those different mission packages in the same hull design, they don't qualify as different ship classes.