I do however think the assertion that the ship was a character on the show to be patently ridiculous. It is an example of fanboy insecurity at its worst. It's an inanimate object within the context of the show/films and a fictitious one in the real world...
It's the same kind of love one would give to, say, a car or old planes and such. Doctor Who would be very different without the TARDIS, the DeLorean is is synonymous with Back to the Future, the Ghostbusters video game has a scene of worry when the Ecto-1 gets incapacitated (also a lengthy "Making Of" section ... for the car!). Some other movies and TV shows go even farther than Trek in this regard: Serenity, Battlestar Galactica, and seaQuest are all the names of the movie/show AND the starring vehicle -- they're not just characters, they're the titular emphasis, to the point where the living, sentient characters worry about the safety of the ship just as much as the safety of the crew (you'll see similar behavior in car documentaries and reality shows).
The point being that it's not just fanboys who equate inanimate objects to characters, because those objects have an injected, almost sentimental value, that's often key to storytelling, because the vehicle is often going to be the thing that ultimately saves our heroes, accompanies them in their adventures, and is very often a literal plot device itself.
Star Trek IV proved that you could have a very successful Trek film without the Enterprise even being present until the last 2 minutes....
The same could be said about Adult Spock in Star Trek III, but that doesn't render Leonard Nimoy any less important or relevant to the franchise. But one could argue (probably a literary critic at that) that having the Enterprise appear in the end was an essential way of ending the film, since it was about tripping up to restore the status quo: Earth is returned to normal, whales are saved from extinction, Kirk gets his classic rank of Captain back (weird that he spent more time as Admiral than Captain and yet everyone thinks of him as Captain, but oh well), our crew manages to stay together instead of being disbanded, Sarek gets his son back from the dead, and finally, the Enterprise returns.