It's important that Clients feel safe within the counselling relationship. If you are sharing your deeply personal information, then you need to know that this is confidential. The counsellor is bound by their professional code of conduct not to share information about who they are seeing, or what is being discussed within the counselling session.
Supervision is a professional safeguard for counsellors to make sure that they are working safely and professionally. I have regular supervision every month, during which I discuss my counselling work, but I do not discuss Client names.
I have a formal complaints procedure should a Client ever feel that they need to use this. For this purpose, I use the COSCA recommended complaints policy. You can read more about this here. COSCA COMPLAINTS