I find your example really funny because of my experience with T. I told T he didn’t really strike me as the pet sort... that is THE longest he has ever hesitated before answering a question! Finally he admitted that he is not and we were able to laugh.
My T is very open with me on the few things I asked about. The one thing he sometimes won’t tell me is where he is going when he goes out of town on business (no problem telling me about vacations or visits to family)... but what makes it extra funny is he will tell me we can’t meet X week because he is going to _____ and he will name the place but then no matter how I ask or how much I pester him he will NOT tell me where he is going when we are in session but will say WHY he is going and this is the only place he goes for that training... even though he already told me previously and knows I don’t forget... then when he gets back he will often say something about going to _____ and use the name again... he hasn’t figured out why I laugh now when he won’t tell me where he is going (and it is always the same place).