Okay, so who "owns" the Doctor? Who "owns" James Bond? Who "owns" Batman? Dracula? Superman?
Well I don't have a horse in this race, but I'd like to point out some things.
If you ask people in general to, say, give you a picture of James Bond, you're likely to end up with a picture of Sean Connery ('iconic') or Daniel Craig (current). Likewise for the doctor you'll get Tom Baker ('iconic') or David Tennant (well-known recent).
The thing with some characters like Batman or Superman or somesuch is that they have a very distinctive appearance. Captain Kirk has none. So what do people imagine when you say "Kirk" ? William Shatner, probably. Now they'll think more and more of Chris Pine, I would hope, but unless the character has a particular look, you associate them with the actor or actress playing them.
Shatner has been Kirk in people's minds for 43 years, so it's normal to assume they'll associate him with the character more than Pine, who's done two movies so far.
My point is that you all have good points. On the one hand, the character is its own thing that often transcends the actors, but you still have people at large associating the two.