But that's your hangup as well as your misunderstanding.
When my father introduced himself as Doctor on the wards of psychiatric hospitals, he was within his rights to do so. And when he introduced himself as Doctor in hospitals, elementary schools, nursing homes and wherever else he was asked to evaluate patients, he was still within his rights and not misrepresenting himself in any way. He was not identifying himself as a physician just by calling himself by his degree: Doctor. In the clinical setting, anyone who has a doctorate can identify him or herself as Doctor, just not as a "physician." Can you enter a patient's room and identify yourself as "Dr. Duluth?" Of course you can, if that's who you are.