Is writing the word "Doctor" instead of numbers on a Muslim physician's car license plate considered a form of showing off or unlawful fame, or is it permissible?

The pivot of seeking fame, showing off (riya'), and self-admiration (ujb) revolves around intention and purpose. If the doctor's intention in writing "Doctor" on his car or house is for fame, self-admiration, or showing off, then it is not permissible. The same applies if his action leads to that outcome. However, if the action is devoid of any forbidden intention and does not lead to something forbidden, such as merely for distinction, then there is no harm.