Serenity doesn't have a show-published registry number, yet she is as strong as a character as the Enterprise.
Albeit one not nearly as closely explored.
Yes, you're right about the registry numbers.
But like the Enterprise D, were were very familiar with the interior of Serenity too. Perhaps it has something to do with the ship being a character... that we feel we know it well, inside and out?
I feel the same way about the TARDIS from Doctor Who, even though they have changed the interior of the console room a few times it has remained the same layout, and always the same exterior shape.
The necessity of the ship too. Has that particular ship saved the crew? Do they rely on it?
You could argue that robots are similar. R2-D2 is basically a dustbin that can't speak, but he is a character as much as any of the rest of the live action characters because he has saved the main characters a number of times, they rely on him, and his shape is instantly recognisable.