"I'd be delighted to, sir. Now that I've got something human to talk about. Rights, sir. Human rights. The bible. The Code of Hammurabi, and of Justinian. Magna Charta. The Constitution of the United States. Fundamental Declarations of the Martian Colonies. The Statutes of Alpha Three. Gentlemen, these documents all speak of rights. Rights of the accused to a trial by his peers, to be represented by counsel, the rights of cross-examination. But, most importantly, the right to be confronted by the witnesses against him, a right to which my client has been denied."
[emphasis added]

Also, the only ship registry stated explicitly was by Kirk:

"United Starship Republic, number one-three-seven-one."

"I'd happily accept the Constellation, the Republic and the Eagle as ships not belonging to the Constitution class, since two of the three were never seen and the third was visibly (if only slightly) different from the Enterprise."

I dunno. The Constellation was pretty close. And considering that the Enterprise changed most of it's details several times an episode (Bridge size, nacelle caps, deflector size, those black bands on the top of the saucer that might or might not have been on the WNMHGB model), and not forgetting that the radically different (by comparison) refit and Enterprise-A were still Constitution-class, I don't think you could claim the Constellation to be different.