There is no paperwork to fill in in my country (maybe if you see a therapist in a hospital? but I'm not sure) so the idea of writing down an emergency contact is odd to me but if a therapist had ever asked me about one, I would have made up a name. Hell, I didn't even give my last name to the therapists I saw (except my ex therapist who also had my address). For some reason they always insisted on having my last name and I never understood why since it's all paid cash and there is no insurance. So instead of pointlessly arguing, I would simply give a fake last name. My last name is really rare and while I don't believe I'm super fascinating I simply don't trust therapists not to blabber to other therapists and to keep my confidentiality. Experience has shown me I was right about that. As for people I talked about, I never used names since it would be odd to refer to, say, David when my therapist has never met that David. So I would just say "my mother" or "my girlfriend" etc. None of the therapists I saw ever asked me about the names of the people I was talking about. I don't see why they would have.