If this helps at all, my T would NOT knowingly see someone one of their clients know. Friend/relative/coworker/whatever. Knowing what my T has told me in the past, if I were in your shoes, I'd talk to my T about how this person approached you about seeing your T, and that you feel weird about it and want to know why. Our situations may be very different, my T is in a large city where there are plenty of others to seek out. And she's not taking in any new clients anyway.

I'm with Brown Owl. I don't tell people, even my family, my T's full name. They know her first name, but again, big city.... I guess my main reason is that I would never want someone contacting her "on my behalf" for any reason, but it is also a relationship that I CAN and am able to keep to myself. I also appreciate that if I'm angry or hurt by her, I can talk to people about it if I choose to and I'm not badmouthing her, because they wouldn't know who she was. I'm very secretive about my T's identity with local friends/family especially....I thought it was quite strange to be this way, but I've told my T about it and she says really it's not all that unusual.