A quick search on the ether produced a 1908 article from the Horseless Age titled INTERVIEWS WITH PHYSICIANS AND OTHER USERS IN VARIOUS CITIES (23 September 1908), only early one so far, that discussed with Doctors the type of automobile they drove for their practice about the time Henry introduced the T. The discussion was on the driving of runabouts. Either gasoline, electric, or petrol. The only early vehicle found so far that had the word Doctor in the name was the 1904 Rodgers Imperial Doctor's Car Rodgers & Co. Columbus, Ohio 1903-1904. By the late 1920s-1930s there were coupes by Duesenbergs, Rolls, Packard, Chrysler and more identified as doctor coupes
As early as 1894 the medical journal Homeopathic News carried an advertisement for horse drawn doctor's coupes.
At A steampunk site there is interest shown in a 1899 question to the origin of a doctor's coupe. http://brassgoggles.co.uk/forum/index.php?topic=38057.0
"……. a 1899 styled Doctor's Coupe Quadricycle. So called Doctor's coupe because they were light nimble city carriages with an all weather enclosed body. Typically bought by doctors so they could perform house calls at any day or hour. Like most automotive terms, the term predates automobiles and was used for horse drawn carriages first."
It's not certain where it got its nickname but some suggests it's because it was popular among Doctor's who found it convenient for carrying their equipment and for traveling to their various appointments. Others suggest it was because of Ford's advertising of the coupe; one of these ads was titled "Dependable as the doctor himself".
So maybe the popularity of a Model T Coupe being a doctors car came from the Ford Advertisement.
An earlier discussion on the forum about the doctor's coupe.
Model T Ford Forum: Forum 2011: Yes, there was a Dr's Coupe