I don't really ask my T a lot of personal questions. He volunteers a lot. I've noticed he never uses the names of anyone in his family, though he does talk about his wife and kids. I don't take it personally though, I imagine it's just a habit and probably there's good reasons for it.

With the situation in the OP--I wonder if the issue might be less about keeping this information private and more about keeping the conversation focused on the client? Like he could say "I have a snake and two cats" or something, but then he's spending your time talking about himself, and there's the chance you'll ask follow up questions which is even more time spent on something other than the client's issues. Same with vacation talk, it seems like it would be easy to end up swapping vacation stories ("I'm going to Paris," "Oh, I love Paris, I went there last year" etc etc.) and lose focus on the actual therapy work. So maybe it's about avoiding chitchat just as much as privacy or security.