Dr. Hook. That's a stage name.
The "Hook" name was inspired by Sawyer's eyepatch and a reference to Captain Hook of the Peter Pan fairy tale, although, humorously, Captain Hook was neither a doctor nor wore an eyepatch. Ray Sawyer had lost his right eye in a near-fatal car crash in Oregon in 1967, and thereafter always wore an eyepatch. The eyepatch led some to believe that Sawyer was 'Dr. Hook'....
You can help me fill out the list of pop culture figures who use the honorific "Dr.," but I'll just note Dr. John...
Dr. Dre...
.
.. and Dr. Demento...
There are a lot of nicknames and pseudonyms in American culture, and "Dr." is a popular component. No need to act like it's sacrosanct and not available for appropriation at whim.