Well even though there was 12 Constitution class vessels it does not mean there was only 12 ships in the fleet. It just means only 12 of THAT class. As you said they are considerd speciel so if so not many will be made or they wont be "special".
In fact, there could be many more Constitution class vessels. The Enterprise of the series proper differs noticeably from it's appearances in both The Cage and Where No Man Has Gone Before. The most reasonable in-universe explanation is that it has had a major refit, increasing the crew size, changing the rear ends of the nacelles, getting rid of the 'rockets' Spock refers to.
So when Kirk says 'there's only twelve like her in the fleet', he might
be referring to that particular configuration, modified for a five year voyage of exploration. Other Constitutions may be configured for other roles.
This is, I freely admit, pure conjecture on my part, with no supporting evidence. It's possible,