Okay, let me try to take a stab at it:
Firstly, one thing I'd want to quickly go over as reference is what it actually means to say that a "stock fantasy setting" is more rooted in the late middle ages/renaissance than in earlier periods. Generally, the traits of these settings that are characteristic of this time are:
Aesthetics generally. Like, castles in a "stock fantasy setting" are usually these big ornamented things covered in towers and turrets (consider, say, Neuschwanstein Castle, which was itself an early modern nostalgia piece rather than a genuine Renaissance fortress), while armor leans more towards full heavy plate (like these 16th-17th century pieces, for instance). These are all classically late styles.
Likewise, your classic fantasy merchant city -- you know, your Ankh-Morporks, your Lankhmars, that sort of thing, with tangled crime-filled alleys, probably a network of scummy canals, a huge bustling port, a guild of thieves, all of it ruled over by dynastic merchants in fine brocade and velvet and heavy jewels -- that's very heavily inspired by the merchant cities of the Italian Renaissance like Venice or Genoa, or North Sea merchant cities like Amsterdam around that time or a little later, or, frankly, even early modern London.
Also, generally, urbanization. Roman civilization was very urban. Renaissance-era European civilization was very urban also, especially in places like Italy, the North Sea, and other areas rich in commerce. In fantasy, too, the notion that cities are major centers of life and politics remains -- merchant cities bustling with commerce, capital cities with the king's fancy castle, fortified cities with huge bloody walls, these are all recurring archetypes. Crack open, say, a D&D setting sourcebook, and that assumption that major cities or city-states are major focuses of culture and economy while rural areas are clearly in their orbits is fairly evident, I think.
Also, in my experience, medieval fantasy monarchies tend to be fairly centralized affairs, with powerful monarchic kings, the royal seat in the capital or the king's castle as the bustling focus of governance, and a tendency for the edicts from the king's person and the intrigues of court to be the driving focuses of political life. Also, a lot of intrigues in a classically corrupt court. This tracks fairly well for the late period, historically, where European kings put a lot of effort into centralizing and strengthening their authority, some more successfully (the Iberian kingdoms at the end of the Reconquista), some less so (Charles I of England). Absolute monarchy was less of a factor in earlier medieval Europe, which operate chiefly under a network of looser, less centralized feudal ties.
By contrast, what would these things equivalents in the earlier Middle Ages be like?
Castle-wise, the really early periods were the time of the motte-and-bailey, aka wooden palisades on top of earthen mounds. Later, you'd tend to see squatter, more fortified constructions for defensive purposes (take the Krak des Chevaliers as an example). Even well into the later middle ages, though, a typical rural keep or castle was often a fortified house or manor more than what we'd picture as a castle today.
Armor was usually dominated by chainmail, scale armor, and non-face-covering helmets (a knight of the Carolongian Empire, for example, might have worn something like a scale or mail shirt and a helm with earguards or a noseguard, but certainly not the "classic" knightly set of plate you usually see in fiction). In terms of helmet styles, for instance, compare this 11th-century nasal helm or these illustrations of 12th-century styles with later forms such as the classic "bucket helm" that was popular during and after the Crusades period. Helmets with more articulated shapes and visors came later. What's called a close or closed helm, for instance, like these ones, was mostly a 16th-17th century style, although antecedent forms were already common in the 15th. Even the barbute with its distinctive T- or Y-shaped visor slit was a distinctly Renaissance Italian thing.
(Like, as a practical example in terms of helmets, this reminded me of a passage in the first Dunk & Egg story where Duncan buys a great helm/bucket helmet, and it's described as coming across as a goofy-looking and antiquated choice when other knights all use visored and often very ornate close helmets. And this makes sense! Westeros is very explicitly based on England during the Wars of the Roses period, in the 15th century, by which point medieval armor and helmet styles had gone out of use and most armored troops wore jointed plate armor and visored helms.)
Religiously, monasteries were often the center of religious life and, for rural communities, often more important than distant bishopric seats. They were also centers of, among other things, record-keeping, writing, and brewing.
Cities. The middle ages proper were a very rural time -- cities existed, and exerted political and economic influence, but took a significant dip in both size and importance for a time. For a good few centuries the center of life for most people was the manorial system, where you had a local knight or lordling living in a fortified house or a small keep and providing protection to a town or two around this in exchange for material support. Self-sustaining religious communities like monasteries were also important. All of these, for a long time, would have basically self-supporting.
(Well, alright, that's also an oversimplification. In rural France-England-Germany, that'd be about right, sure. In northern Italy and in Flanders-Saxony, though, you start to see... not urbanization as such, but, let's say, townization early on in the system of the medieval communes, where farming and trading towns left isolated by the collapse of Roman-era state systems formed confederations for mutual defense. So the main form of urban culture in this timeframe isn't really centralized around a single big city but more along the lines of fifty towns who formed a loose alliance because they're afraid of bandits and robber knights.)
Government. Centralized states and nations weren't... really things for a long time. Feudal governance was based much more strictly on the notion of personal debts and obligation between rulers and those who administered portions of land for them. A feudal noble had obligations towards his king, but he also had a tremendous amount of freedom in running his own affairs and could have gone for the majority of his life without every showing up in the vicinity of his lord's castle -- the idea of, like, the royal court as the focus and center of political life wouldn't really crystallize for some time, and this degree of independence meant that negotiation and re-negotiation (read:testing how much you could ignore the king or going into outright revolt) were very common.
Also, the notion of national identity -- of saying "I am French, I am English, I am Italian" -- also didn't really exist (I'd recommend Liah Greenfield's Nationalism: Five Roads to Modernity for a discussion of how this mode of thinking about group-and-land identity formed in the 16th-17th centuries). Before then, concepts of identity were either strictly local ("I am from this village"), based on loyalty systems ("I am from a village defended by Knight Whatshisname, who gives his tithe to Lord Whatshisface, who gives his tithe to King Whatshisproblem"), or based on religion ("I am Christian", "I am Jewish", "I am Muslim").
So, broadly, there weren't really "countries" the way that we think of them today, with clearly delineated cultures and borders and directed from a central capital. The most granular political units were the rural manorial keep and the free town, which gave their fealty to a larger lord or a grouping of peers, and the span and size of a kingdom or an empire was determined in large part by how many of these tiny units or groupings of tiny units you could claim tithe-right and allegiance from. And that's feudalism!
Event-wise, the Big Things in the medieval period proper were the Crusades, a series of primarily religion-motivated wars (or, well, religion-land-and-luchre-motivated wars, granted) and the Black Death. It really cannot be overemphasized how the terror of being struck by this mysterious dying dominated medieval thought for large spans of time.
One book I read recently which I'd recommend for a having a good look at how medieval people thought and lived might be Michael Flynn's Eifelheim... with the caveat that it's also a science fiction novel about first contact with aliens, but the medieval side of things is really well-researched and -written.
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