The reason he does not have Dr. Fantastic in his name has to do with his origins at the hands of Jack Kirby and Stan Lee.
According to Stan Lee, “I was utterly determined to have a superhero series without any secret identities. I knew for a fact that if I possessed a super power I’d never keep it secret. I'm too much of a show off. So why should our fictional friends be any different? Accepting this premise, it was also natural to forego the use of costumes. If our heroes were to live in the real world, let them dress like real people.” -- Secret Origins of the Fantastic Four
When creating the Fantastic Four, their original premise was they would be Everyman heroes, no secret identities, only codenames. So the day they acquired their powers after the accident, their identities chosen immediately were: Mr. Fantastic, the Invisible Girl, The Human Torch, and the hideous Thing.
Stan Lee would only be able to keep his word against costumes because the fans demanded they wear them! By issue three, they were issued their non-uniform uniforms and their codenames were firmly entrenched into history.
There was one other reason he remained Mr. Fantastic. The name of his hated arch-enemy, one Dr. Victor Von Doom. Kirby and Lee wanted to be sure to distinguish between the two mentalities of their creations. Richards was allied with the common man, using science to further the opportunities of mankind. Doom only had one ambition, to rule mankind and thus did everything he could to separate him from mankind, making him hideous in appearance, a member of royalty, indeed the ruler of his kingdom of Latveria and a firm belief in his superiority over everyone but especially Reed Richards.
NOTE: Dr. Reed Richards (Mr Fantastic) indeed has multiple degrees in a number of sciences and that number grows every decade. It is safe to say if he needs to know it, he does, and if he doesn't he maintains company with the worlds most famous and capable scientists and inventors including Tony Stark, Dr. Henry Pym, Dr. Henry McCoy, Dr. Charles Xavier (deceased), Dr. Stephen Strange, and even Victor von Doom, (who left school before he could get his doctorate) if the stakes are high enough.
Read the original outline notes for the first issue of Fantastic Four!