There is a minor controversy brewing over whether Joe Biden’s wife Jill should use the title “Dr.” It is an issue that confused my college students as well. And it is a problem, for the word has multiple meanings.
If a person has earned the highest degree in their field of study, they are entitled to use the term “Dr.” in place of “Mr.” or “Ms.” with their name. But not without their name, unless their field of study is medicine.
When we say “Doctor, what do you think?” we are addressing a physician or other health professional. When students addressed me, the appropriate form was and is “Professor, what do you think?” or "Dr. Cotton, what do you think?" Used without the name the title specifies ones field of endeavor, not that one has the so-called “terminal degree.”
Journalists frequently get this wrong. I remember reporters asking Dr. Henry Kissinger questions like “Doctor, what do you believe the North Vietnamese seek in these negotiations?”
Doing so, they revealed their ignorance. When he was SecState the appropriate usage would have been “Mr. Secretary” or alternatively, “Dr. Kissinger.”
Used without the name it becomes a job title and only fits those in medicine, and perhaps optometry and dentistry which are also health care. Used with the name it indicates a person of high educational achievement in any field in which the doctorate is given. Historically some fields like Art and Music have not offered a doctorate, the Master of Fine Arts (MFA) is normally their highest degree.
The quick-and-dirty way to know when you can call someone “doctor” without adding their family name is if the answer to the question “What does he or she do for a living?” is “He or she is a physician.” If the answer is “He or she is something other than a medical professional with doctorate." the answer is no.
Summary: Addressing Joe Biden’s wife as “Dr. Biden” is appropriate. Without the family name, one should say “professor” or "instructor," whatever faculty are called at the community college where she teaches.