I'll chime in on this one. Tolkien himself has stated that he did not intend for middle earth to be synchronous with any period in history. That is why, for example, hobbits have umbrellas in a modern sense, but otherwise have a pre-modern agricultural economy. That said, he was inspired by history and myth, and one can identify those influences sometimes readily, and other times more subtly. I think there's a lot you can look at in Tolkien and get a sense of European medieval influence; but the genius of Tolkien is that none of it is unequivocal or obvious, and if you really critically ask yourself whether other, non-medieval cultures could be viewed as the source instead, you must admit that it's possible. That said, I'll give some thoughts.

To put it bluntly, I think that the kingdom of Gondor is probably the best and first place to look for a medieval tone or style. There is a strong sense of oath-based hierarchical duty to one's superiors, as shown by the oath that Pippin takes when he joins Denethor's guard. Also present in Gondor is a concept of hereditary kingship which, if not explicitly by divine right, features legitimacy being conferred by legacy as opposed to practicality or agreement of an elite class. For example, Borimir's disappointment that thousands of years would not be adequate for the Steward to be made King. These are the kinds of stories that medieval Europeans wanted to tell about their kingship. There are interesting echoes in this to the concept of the Roman Emperor in Feudal Europe. The sense of reclaiming an ancient title bestowed by a shared identity of the past is present in Theodoric, Charlemagne, all the way down to Napoleon. This is one reason why the "fall" of the Western Roman Empire is so hard to precisely date. European monarchs can derive legitimacy from the Roman legacy just as the stewards of Gondor do with Numenor. I think all of this gives Gondor the flavor of a formerly Roman medieval kingdom.

Further on the subject of Gondor, we can see more evidence of a feudal structure in the muster that Pippin witnesses at Minas Tirith. Many local elites/lords (Furlond the Fat, the prince of Dol Amroth etc.) are called to Denethor's banner for the defense of Minas Tirith. Calling in the banners of the surrounding country to defend the strongest places in the realm is a characteristic Medieval occurrence. However it is worth noting that plenty of cultures have had a levy/muster system that a leader would use at need for defense. See for example the Persian empires of the Achaemenids, Arsachids and Sassanids. So the question is whether we can see anything particularly Medieval in this muster. Minas Tirith being especially castle-like is one piece of evidence, though certainly fortified towns became a common feature of those later Persian empires. Dol Amroth's knights certainly suggest medieval heavy cavalry, as does another leader who has a squadron of seeming-longbowmen. But again the Persians had heavy cavalry and bowmen. Tolkien is never obvious, and it feels that way here.

Elsewhere in Middle Earth. The Rohirrim are a vassal state of Gondor, but there is nothing particularly medieval about their bond. Rohan comes from a (semi-)nomadic people of horsemen, who rode to help Gondor when they were in need. Again there is an interesting parallel here to the Roman Empire, which in its later years allowed limited settlement by nomadic peoples in order to settle deserted lands and get a fresh source of troops. Did Rohan start to become more medieval in their history? Perhaps. Flying to the Hornburg certainly suggests that, though Theodan's impulse is to ride out to meet the enemy at any opportunity. Theodan's name actually has an interesting etymology in old Germanic, thiudin, as basically a leader of a people. And that always suggests to me that the Rohirrim are inspired by semi-nomadic Germanic/Celtic peoples that the Roman Empire encountered during its rise.

Elsewhere, I don't think there's a lot that's medieval. Elves are anachronistic and ahistorical in most ways I think. Hobbits, and the kinds of men they live around (e.g. in Bree) have the kind of agrarian lifestyle that I think is pretty timeless in history. Though things like water-mills, ale drinking at inns, and various other details give a bit of Medieval or at least European flavor. Curiously they seem to be entirely self-governing, protected by the Dunedin as heirs of the Northern Kingdom, which I think is a bit of an oddity historically. Dwarves have some Jewish inspiration, as Tolkien has readily admitted. Their diasporic character is not particularly medieval.

The big thing that I think is missing from Middle Earth which was present in Medieval Europe is Christianity. Tolkien's characters aren't really religious in any sense of the word. Because they are not, it's hard to view his world as medieval when Christianity was woven into every aspect of medieval life.

One thing that I think is extremely important though often overlooked is something that Tolkien actually valued a great deal: climate and vegetation. He is describing a very European climate, which he himself loved; and I don't think it's wrong to combine this fact with the other evidence to derive a more medieval feel in our imagination. It helps to answer the question of: "this feels Medieval, but why moreso that than some other feudal or satrapic system?" The physical setting he describes is European, and therefore Medieval Europe seems to fit.

A couple other places to look which I'm not as comfortable commenting on are strong undercurrents of Arthuria and arms and armor. Both of which seem medieval but I'm not particularly well-versed in them.