I was watching 'Faster' the MotoGP documentary. It is a little Rossi heavy and they talk about this.
Rossi says he kinda picked it for himself because Doctor sounds so respectable. "Ohh.. you go to towna and the doctorr he is respectad by everrywon"
Words to that effect. I think it was Rossi being a goofball. Some how he wasn't seeing himself as a racing legend, a huge celebrity etc etc. Which I suppose he wasn't when he picked his own nickname. This really contrasts with how Biaggi probably felt at the same time. He thought himself as being a big shot, a fancy motorcycle racer, dating supermodels, being rich etc. This is part conjecture of course, but they tried to portray that a bit in the documentary.
5. Rossi has an honorary Doctorate in Communications from the University of Urbino
and thus became the DOCTOR. OFFICIALLY