No, not all mental health counsellors are doctors. There are different counsellors and here is a list which will help you in understanding the counsellors better:
Psychologist/Counsellor: A professional with MA or MSc degree in clinical or counselling psychology. These are equipped to handle cases like anxiety, depression, marital or relationship conflicts, self-issues, etc.
Clinical psychologist: A professional with M.Phil. degree in Clinical psychology. These are equipped to handle the psychotic cases like schizophrenia, bipolar disorder etc, along with other cases like anxiety, depression etc.
Psychiatrist: These are professionals who are doctors. Only they are allowed to prescribe medications for any mental health conditions.