Yes, mine is like this too, you have my sympathies. I've been seeing him for over 10 years and find it incredibly painful that if I ask him a question about himself, he won't answer it. This way of working has impeded connection and trust within our relationship. I coped by excessively and sometimes obsessively googling him for more information. I wouldn't recommend my coping strategy If you can talk to him about how you feel and try to explain, that might help. I wonder if he would give you an explanation as to why he won't answer?

You might need to decide whether it's important enough to you, to find a new therapist who works in a different way.