Do you care about your house? You built it yourself, so obviously you took pride in your work and made it special just for you and/or your family, yes? It works the way you want it to work, yes? And you would be sad if it was destroyed, yes? And living in your house is preferable to living out in the street, yes?
No, it won't miss you when you're gone, just like the Enterprise wouldn't miss Kirk or Picard if they left it. That's not the point. Nobody is trying to anthropomorphize your house, or a particular hero ship. The point is that the audience should care about the ship almost as much as they care about the characters. Because if you don't care about the ship, then what's the point of having it in the first place? The characters might as well just beam to each planet-of-the-week instead. I remember seeing the Enterprise blowing up over the Genesis planet on the big screen in STIII, and it was an emotional experience, just like it was when Spock died. And I remember seeing the Enterprise-D blowing up over Veridian III and recall being extremely pissed off that this ship that I had been invested in for the last seven years was destroyed in the stupidest way possible. Just like how annoyed I felt when Trip Tucker died in the stupidest way possible. The only difference in those examples is that one thing is an object, and the other thing is a person. That doesn't change the similarity in which I viewed those things. If I didn't treat the ships like characters, then I wouldn't have cared that they got destroyed, no matter how it happened. But I did care, because they are just as integral to the story as the people are.