I agree that "Do you have any pets?" can be understood as an invitation for the other person to talk about their pets in greater detail. But as someone said above, this is therapy and not casual conversation. The therapist might be trying to focus on the client, rather than deliberately withholding information. Of course, LT could just ask her T why he didn't answer the question more fully.

I'm reminded of the scene in Parks and Rec when Leslie meets with a political consultant and asks some normal small talk question, and the consultant says something like "You're paying me $200 for this hour, do you really want to spend that time talking about the weather?"