Working with the above factors constituting what is "medieval," let me see what makes sense to me:
Out of Feudalism, Castles, Mounted Warfare, Cities, Royal Power, and A Merchant Class, D&D neither implies nor denies most of these. I think Mounted Warfare is in D&D (the feats), and there has been a lot of new stuff on cities (and by cities, we mean significant urban centers-- as in, more than just a big fort with stuff around it). And say what you will about the "magic item economy," it definitely implies a strong class of merchants. But there is no social system implied in D&D, at least none that is feudal or involves royalty, and there aren't many rules on castles.
Now, what is decidedly not medieval? Lots. Plate armor is pretty much post-medieval, as is the rapier. Druids were on their way out as soon as Caesar hit the scene-- although, expanding the definition a bit, there were still pagans the Christians had not conquered well into the High Middle Ages. That said, monotheism dominated Europe during this time, so the traditional cleric ain't too medieval-- although they do have a bit of a Hospitaller/Templar feel because of their martial nature. I suppose the barbarian is just anybody from outside the whole Euro-Christian thing. The spellcasters... well, people believed in witchcraft, and since magic isn't real (sorry folks), let's just accept sorcerers/wizards as medieval cognates of an eternal archetype.
I hope we can all agree the monk ain't medieval anything?
Maybe it is best to leave the mystical elements alone (turning undead-- where did Gygax get that from? Dracula?), since most does not have a strict historical analog.
Stuff like the double-sword is of no time period. Ditto the repeating crossbow, which is kind of in that clockwork/steampunk vein. Or something.
Now, I am only talking rules. The art direction... I got nothing. It is what it is. It sells books.
As for how medieval D&D should be... well, since D&D is a fantasy game, fantasy fiction is a much bigger source for it than history. Logically, though, you and I know fantasy owes a lot to real world history and historical folklore. So, how medieval are the sources of D&D?
Tolkien? Lots of forts, many castle-ish. Mounted warfare, though it mostly reads like vikings on horses and not real-world cavalry. There are cities and kings. There does seem to be a merchant class, but only around the Shire. A lot of talk about armor, but I don't remember too much specifics, so I don't know if there is plate. Religion is much less prominent on Middle Earth than it was in Middle Europe.
Dying Earth? Set in the future, it's hard to say. I remember liking it, but as I write this all I can recall now is everything being "prismatic." Heck, it was really just the model for the spell system, let's move on...
Elric? Sort of meant to be prehistoric. Basically, everyone is a slave of Melnibone or they're not. There are some kings and titles. The tech (weapons and armor) is decidedly medieval. Not castles as we know them, I don't see much of a merchant class. Mounted warfare, there is some. there are certainly cities. A lot of polytheism.
Lankhmar? Lots of cities and merchants, but pretty low on feudalism. It ain't about knights or kings, but it has a gritty feel some of us associate with that medieval. The tech here is medieval too. Polytheism, but not of any period-- the gods in/of/around Lankhmar are just plain weird-- Religion works a lot like D&D.
Conan? Never meant to be medieval. Medieval elements, sure, but really just prehistoric stuff.
So, is D&D medieval? Well, there's not much implied socially by D&D itself, and a lot of defining the period seems to be about social systems. D&D does imply polytheism, which is decidedly non-medieval. A lot of medieval elements are in there (like someone above pointed out familiars), but it is alongside a lot of renaissance stuff (plate and the rapier being the big ones). It seems to imply that in default D&D there are remnants of the world's medieval past, but things have moved on to be more like the Renaissance.
Then again, there's a lot of stuff that is not of any Earth time period, like the Repeating Crossbow. Oh, and obviously magic and other races (like elves, who use rapiers), which never existed, though these elements are assembled from folklore all over the place.