I very rarely asked my Ts any personal questions about themselves, in fact I can't easily recall a single personal question to my first T (he puts a great deal on the public web so I knew more than enough). Second T disclosed quite a lot on his own and I never had the impression he was holding anything back on purpose. He talked about his pets as well.

Your T may not have a very clear philosophy on how he wants to handle self-disclosure, what sorts of things would not be revealed and what is okay... maybe he just handles it as he feels in the moment, without setting solid rules for himself. Isn't he somewhat unpredictable with some other things as well, including when he will respond to an email?