Why do you assume he's not licensed? Everybody's gotta have a fallback career, and with all those coins he's collected over the years he could definitely afford medical school.
He spends plenty of time rescuing princesses, playing sports, and riding dirty to have time to attend medical school. Even if he did, he prescribed way too many pills to be an effective doctor. It's as if he isn't even trying.
There's also that theory that Mario buys immortality at the cost of 100 collected coins, which quite frankly is a steal to live forever. We never really see him bank those coins. He still lives in a shack with his brother as a roommate.
It is the weirdest choice. The only thing I can think of for it is that a character just wasn't working and got cut and this was the fastest last minute replacement they could do.
That makes sense and probably what happened. However, they could have cloned Mario and given us Toad. Mario and Dr. Mario are the same character. Dark Pit, for example, is a literal clone of Pit in Kid Icarus Uprising. He could and should have been an alt, but at least he's an actual separate character. Even Link and Toon Link are technically separate. They're different Links. The priorities are all over the place. It's so strange. Personally, I would have rather seen the time spent rebalancing say, Dark Pit used to give Toon Link some different moves. Make him closer to Luigi than Dr. Mario, some similar attacks and movements, but they're ultimately very different.
Additionally, the inclusion of custom movesets makes the existence of clones all the more baffling.
So if Wolf and Lucas aren't in the final version, even as DLC, I wouldn't be surprised if Dr. Mario, Dark Pit and Lucina were Sakurai way to remind everyone who complained about Wolf and Lucas what real clones look like again.
That sounds extremely petty. Make the game worse because people on the Internet are weird? Fine, I guess. Sakurai is the same person who pouted about people watching the Subspace Emissary cutscenes on youtube so I suppose this is possible.