As other people have said. The only thing I would add is, depending upon the location of the office and how well-known you are, someone could recognize you in the waiting room. I can speak from personal experience on that one too (as someone who recognized a person in public office).

He was actually a client of my therapist's and my therapist had made vague references to him a time or two as a teaching moment for me, even mentioning that he told the guy someone might recognize him in the waiting room, since he was in politics of a nearby town.

At the time, I had no idea who he was (didn't think I'd recognize him either, even if someone pointed him out in a lineup). But sure enough, one week our times lined up. He had the slot before me and I was waiting for my turn. When he came out and walked through the waiting area, I immediately recognized him (because he was in politics for quite awhile in my home town).

I never said anything to him 'cause 1) it's none of my business and 2) good for him to be getting help for whatever; it's the only way to work on destigmatizing mental health.

At the end of the day, I think the likelihood of someone recognizing you, unless you're really up there, is pretty low. The only reason I did was two-fold luck: that I grew up in that specific town, which I doubt 99% of people going to that practice did and our appointment times crossing so I could even see his face