I just think a therapist being all secretive about it makes it a much bigger deal than it actually is. When a therapist refuses to answer a simple casual question, I think it is weird and kind of mean. For instance if I am talking about my cat and I ask "do you have any pets?" I think it is much better for them to say "yes, I have a dog," than to say something like "I don't think that's relevant." Now if a client is constantly barraging the therapist with personal questions, then I think that's different.

Personally I try to avoid asking very many personal questions at all because I've had therapists who follow up the casual question about pets (which I really only asked to be polite) with a 10 minute discussion of her wonderful dog, which is the other end of the self-disclosure spectrum.