Short answer:

The most literal definition of "medieval" is "between two ages". Taken that way, the term might be considered "Roman centric" ,"Mediterranean-centric", and perhaps even "Asia [Minor] centric" -- it refers to the time between the fall of Western European Empire and the fall of the Eastern Roman Empire -- most of which was in Asia, not Europe. But the reason the term continues to be used is because it describes historical dynamics found in many places.

Discussion:

François-Xavier Fauvelle, an historian and archaeologist of Africa, entitled a recent book "The Golden Rhinoceros: Histories of the African Middle Ages". Why did he refer to an "African Middle Ages"? Does the fall of Rome or the fall of Constantinople have anything much to do with African history? Not really.

In his introduction, he explains his choice of the term:

[L]et us admit that there can be many reasons to use the term “Middle Ages” or the adjective “medieval” that are not particularly related to the way medieval Europe is medieval. There’s also a good reason not to use it; for if its usefulness resides only in designating a period of almost a millennium roughly coeval with the European Middle Ages, one could rightfully ask why we should import a label that conveys unwanted associations with medieval Europe: Christianity, feudalism, the crusades against Islam. True. But despite all this, I think that applying the term “Middle Ages” to Africa is justified.

The justification concerns the scale at which we observe the Middle Ages {snip}

[T]he physical centrality of the Islamic civilization within this global world, the role of specialized long-distance merchants (mostly Muslims and Jews) as connecting agents between different provinces, or the related significance of a few chosen commodities (such as slaves, gold, china, glass beads, ambergris) [is] evidence of an interconnectedness of a kind limited to what met the needs and tastes of the elites. In that sense, the broad picture that this book wants to draw, its fragmentary nature notwithstanding, is that Africa also deserves to be considered a province of the medieval world. Not out of a will to “provincialize” Africa in the sense of making it marginalized or peripheral, but, on the contrary, to make it part of a world made up of other such provinces.

Fauvelle thus claims a case for the generality of the term "medieval", if we discard its literality.

Literally, "medieval" means "between two ages". No one who lived at the time could have made sense of that. They might have known -- and indeed lamented-- the fall of Rome. Christians might have thought of themselves as being "between" the appearances of Christ on Earth, but no one could have understood the periodization that the later Italian humanists like Petrarch applied, between the loss of Rome, and the rediscovery of Rome.

So we leave aside the Merriam Webster definition; its literal, but can't explain why anyone would use the term today. Why does Fauvelle speak of an African "Middle Ages"? Why did Gustav von Gruenebaum speak of "Medieval Islam"?

Fauvelle gives us a clue as to one reason-- it's an age when earlier trade relations have largely collapsed, and knowledge and goods move through the world transported by new agents, Jews and Muslims for example. In the Africa he's describing, the historical record is almost entirely recorded by Muslim traders, soldiers, missionaries and explorers. Sub-Saharan Africans leave us very little written material from the period; but the institutions of travel in Islam - particularly the hajj -- mean that not only do we have records of Muslim trade with Mali, we have the pilgrimage of the King of Mali, Mansa Musa, to Mecca.

Seen in this light, the "Middle Ages" is a period after antiquity in which the religions we know in the modern world assume roughly their current geographic distribution, and this isn't Eurocentric. Indeed while Christianity may be thought of as a "European" religion-- it is in fact an Asian religion which "conquers" Europe. Similarly Islam is an Arabian religion which explodes into Africa, Asia, Europe. And Buddhism, which had begun it spread a bit earlier, comes to dominate China, Japan and Korea . . . The Tang Dynasty is often characterized as the age of "classical Buddhism" in China.

Considered as a practical matter, the period "medieval" tells you something about the languages you might need to know to study history. A "medieval historian" of almost anyplace in the Mediterranean world will often need to read Arabic, Latin and probably Greek in addition to other languages. If you're an historian of 17th century Spain-- you don't particularly need to read Arabic or Hebrew; if you're an historian of 12th century Spain, you do. Similarly a student of medieval Islam may well want to read Greek or Latin, Hebrew, Coptic or Italian, in addition to Arabic, Turkish and Persian. In the same vein, if you're a scholar of "medieval" Indonesia-- you're often going to want to read some Arabic to understand the history of the spread of Islam.

So this is a period where parallel dynamics of religious and political spread can be seen around the world. The arrival of Islam in what is now Pakistan, the arrival of Christianity in Scandinavia-- there are commonalities here. This is a period in which peoples adhere to a world civilization, which shares ideas across thousands of miles. Scholars in Cordoba and Damascus read the same documents and argue fine points of theology and jurisprudence; scholars in Ireland and Constantinople do the same.

Seen in this light, the "middle ages" gets another meaning-- its the time of the spread of world religions, the time in which a previously largely polytheistic world is adopting monotheism, and religions that are heavily based on sacred texts, which require literacy and exegesis to apply. Greeks and Romans argued about "what is a good life" -- sometimes with a reference to "the Gods", but never with a deeply textual codified scholarship. What we see in the "Middle Ages" is the rise of a new scholarship, scholars who engage the scripture to derive rules for living that apply to tens of millions of people-- and which still do today, in the Americas, Africa, Europe and Asia. Jews in particular had developed this style of exegesis and and application to daily life earlier-- but "the Middle Ages" is the period where what had been a kind of analysis and scripture driven policy moves from the practice of a minority religion, geographically quite limited, to the practice of much of the literate world.

Finally, it's a common characteristic of this period that religious doctrine becomes intertwined with political interests in a new way. Looking back to the world of antiquity, we don't have much evidence for religious doctrine as a twin to political interest. Athena was the tutelary deity of the Athenians, but we don't get any sense that they fought the Peloponnesian War to get others to erect statues of Athena; when they put the boot in on Melos, that's not one of their demands. But Christians and Muslims alike -- as they spread their faiths, they have a demand "tear down your temples to your old Gods, ours isn't interested in company".