Mario studied medicine in college, but decided to become a carpenter instead (before switching to plumbing).
Shigeru Miyamoto has hinted that the Dr. Mario games don't count as part of the (implied) canon of the Mario series. It could be said however, that Mario, and maybe even Luigi might have studied medicine when they were in college, but simply decided not to become doctors. Admittedly though, this doesn't explain Princess "Toadstool"note being a nurse (according to the instruction manual for the U.S. NES version). The Dr. Mario series takes place within an Alternate Continuity, completely separated from the mainline Mario games.
In Dr. Mario's universe, Bowser never rose to power or was defeated for good. This might explain why we never see Bowser in any of the Dr. Mario games. Since Peach was never or stopped getting kidnapped, Mario didn't have a need to venture out to save her. He decided to take up medicine to fill his time.- Jossed, since Bowser and his troops appear in Dr. Mario World.
Mario is simply dressed as a doctor as a part of a plan to get rid of the viruses, which are monsters, not actual viruses.
It's been established that "Dr." Mario isn't a real doctor, and Mario isn't the sort to perform back alley medicine, so, clearly, something else has to be going on here. The Mario & Luigi series shows that viruses are monsters, not literal viruses, though they may be related to actual viruses. There was an outbreak somewhere, and Mario disguised himself as a doctor and used the "pills" to exterminate them. Wario created the viruses.
Assuming Dr. Wario is canon, Wario also became a doctor and wanted to make tons of dough, so he bioengineered the viruses to draw patients to his clinic. The Viruses aren't actually malevolent.
They're just upset that they get left out of all the go-karting, golf, tennis, board games and such that so many other Mario characters take part in. They're lashing out, and it seems kind of justified since no one remembers them come time to have fun, but the entire main cast can gang up on them when they go play the only way they know how.This isn't actually Mario Mario
It's another member of the extended Mario family (probably not another brother) who happened to become a doctor.