Whether she told him her name or not is irrelevant, it’s crossing a line and exposing info about you that he doesn’t need to know, particularly without your consent.
Having been burnt and aware of people with a grudge who like to talk about other providers exposing information not meant for my ears or anyone else, I’d certainly address it on a serious note and be very careful about what else is shared with this person. Might be irrelevant to her but it could be costly to you. If your name slips out, what else could be shared?
I’ve had a client find my name from picking up my post in the hall on his way for a shower, he used my real name in AW feedback and would use it in comms which I hated as I never wanted him to know. I did address it and say it made me uncomfortable and I wanted to stick only with my work name but my name kept slipping out his mouth so let’s just say he’s no longer a client.
The key thing here is, this was done without your consent and if it makes you uncomfortable, you’re more than entitled to be annoyed or to not see this client or “friend” again.