My Ts family doesn't have much of an internet presence (which I can see), the most would probably be my T. But I have both grown up with researching people online and there is some stuff on the internet of them, so I do have some experience with this. Though to be honest I have never talked to my T about it, mainly because I do not feel bad about it.

My T has a unique name (in my country). I know his wife's name because she works in the same office, though for her she literally has no internet presence at all. From his name, as well as seeing his license plate once I have figured out:

- his address
- his children's names
- one hobby of his girl and one of his boy
- photos and other data connected to the hobbies, as well as some photos of himself
- the school his children go to

My thoughts on this are the following: although some people do not consider it or sometimes I feel like aren't even aware of it, things you put on the internet are public. Never, ever assume that anything you put on it will disappear, nor that it could not be seen by literally anyone on Earth. Not because the internet is some magical place, but because as soon as something is online, somebody might have seen it, and they might share it, use it, publish it somewhere else without any of the precautions you took or without care for the fact that you want it deleted. And this just snowballs. Of course this doesn't always happen, but it can happen.

So, the people you look up decide themselves to put up stuff for the whole world to see. It's certainly not intrusive to see it.

Now, to me I do not obsessively check these things, I have saved everything I can and if I need a certain kind of comfort, I might access it. If you feel like you struggle with looking things up too much, that's certainly something to look at and discuss in therapy.
I also would think it to be problematic if I now showed up at his address (without there being some kind of unrelated reason at least), if I went to his kid's training to see his kids, or something like that.
A good example is there is the real possibility that I walk into his son travelling to military training where he'd have his last name clearly displayed for anyone to see. I'd obviously look at him. I'd obviously not go up to him, say hi, your dad is my therapist, so how is he in real life. I'd probably tell my therapist about it just to share, but I'd not feel ashamed of it.