One thing that clients often misunderstand about confidentiality is that it is essentially one way. It is unethical for therapists to reveal confidential information about a client without the proper consent or appropriate reason. But there is nothing that goes the other way. Your friend may or may not approve of you talking about him with a therapist he is seeing, but it would stand to reason that he might expect that you would do so since he referred you to his therapist. It is likely that he has talked to her about you.

Secrets in therapy, just as in any relationship, stand in the way of healing. I can see a legitimate purpose in establishing who your friend is. You knew, and your friend knew, that you were both going to the same therapist. How much chance is there that she wouldn't have put together who is who? I'm betting that she knew and had no doubt who referred you, and her purpose was to get that bit of information out on the table in order to prevent it from being an awkward secret that could interfere with your therapy. She did not reveal any confidential information, because she did not tell you anything that you didn't already know, or that she didn't already know that you knew. The only thing revealed was the fact that she knew (or had intuited) something that you didn't know that she knew.

You now have options. You could talk to your friend about what happened, thus establishing that your identities are out in the open. Then you could find out how okay he is with knowing that the therapist knows who you both are. Maybe he will be okay with that.

You could talk to the therapist about your concerns and ask to clarify how confidentiality applies to your situation. This will give you the chance to see if her explanation is acceptable to you, and decide whether or not you can feel confident that she is ethical.

Knowing that she knows your friend, you might benefit from additional insight that the therapist will have from knowing the other side of the story.

If you and your friend decide to, you also have the option of an occasional joint session if it would meet your needs.

Of course, deciding not to continue is also an option. But I would get more information first before making that decision. I'm not sure that any ethics breach has occurred here, and this might be a therapist in a good position to help you,