It's in the therapist's best interests to not reveal much, personally or professionally, because it keeps them out of trouble and keeps the client guessing.

But the more opaque the therapist, the less basis there is for trust. And that burden is 100% on the client.

The client is expected to trust a stranger, and surrender control to an all-knowing parental or god-like figure who dribbles out a few disclosures here and there..

Seems to me the question about pets isn't just about pets, it's about figuring out who you are dealing with based on the way the conversation goes.