In my opinion, you should talk to her about it.

I am like this with people, though luckily not with my T. I can imagine how bothering this must be. After years of struggling with this, I finally realized that it's probably due to severe attachment problems, and that's something that won't just go away on its own. And since it's very clearly bothering you, and Ts are there to help you with things that bother you, she would be a very good starting point as someone to talk to.

Now I understand that you don't just wanna go "hey, I stalk you and your whole family", that might freak her out. How about just talking about it without mentioning any names? From my experience, the questions that come back won't be "who is that person?", because really, that doesn't matter a whole lot. You two can still work on skills and things you can do to help yourself.

And if things then don't get better, and you feel that it really only could be solved if she actually knew it was her, then you can still think about mentioning to her that it's her. She is a professional, and she should be used to the fact that her clients might look her up, and that depending on their issues that might escalate. She should (in theory) be able to deal with it in a professional manner, without hurting you or freaking out herself.