I don’t get the double standard argument. By that same logic, you go to the doctor and tell them you have symptoms x y z and man my back is killing me. Doctor listens to you then tells you she has pains a b c and oh boy these knees of mine are a mess. So when I go to therapy to talk about my abusive ex, the therapist should also talk about hers.

By the way, here’s what the American Psychological Association thinks of the whole thing:

“How about Googling clients — should you?

In certain circumstances, there may be a good reason to do a search of a client — there may be an issue of safety, for example. In certain kinds of assessments, it might be a matter of confirming information. But again, we always need to think about how this fits into the professional relationship, and what type of informed consent we’ve obtained. Curiosity about a client is not a clinically appropriate reason to do an Internet search. Let’s put it this way: If you know that your client plays in a soccer league, it would be a little odd if on Saturday afternoon you drove by the game to see how your client is doing. In the same way, if you’re doing a search, thinking, “What can I find out about this person?” that raises questions about the psychologist’s motives.”

The Internet’s ethical challenges

They maintain that stance elsewhere on their website.

So the OP’s concern seems completely valid to me and the therapist’s poor response also seems reason for concern.