How is the term "medieval" used to describe non-European civilizations and cultures?

Simple, because roughly after the Middle Ages comes the Navigations which integrated the world, and a visible European dominance in some important issues.

The history of Asia can not ignore Europe after the XVI c. Before that, Europeans had to travel by land, as Marco Polo and the Franciscans who reached China, and the Muslims separated Europe and East Asia. Thus, Europeans had a relatively smaller influence. Asian story was more localized and independent.

After the navigations, Japan traded and closed itself, India traded and started to be colonized, China had Portuguese in Macau and Mateo Ricci. Spain got into Philippines, Portuguese in East Africa, India, South East Asia, etc. The Ottomans and Muslims were bypassed in trade with loss of wealth. Besides, all of them had their own complex relations with western tech, ships, science, astronomy, firearms, weapons, religion. And obviously this kind of global issues never really stopped.

Studying medieval Asia is relatively more limited to Asian concerns, post-medieval Asia has to take more European and global issues into account. This means that it makes sense to separate the periods when studying history - this is what makes these "period names" relevant or not, even if one chooses different exact dates to mark the transition in different places. These periods are just meant to help our understanding - nobody in the XVI c. had a party to mark the end of the Middle Ages :-)

e.g., if pre-history means 'before writing', this is helpful because historical methods must be different if there is no hope of finding written records. Some may prefer that "pre-history" mean 'before agriculture', it also makes sense, as hunter-gatherer societies are different from settled ones. Separating Modern Age from Contemporary Age in the French Revolution makes sense if one is concerned with political history, about monarchies x republics, and related concepts about human rights and the legitimacy of the state, etc.

EDIT about China/Japan being closed:

there were significant numbers of Christians in China and Japan before persecutions; Jesuits were even present in the imperial and shogunate courts, got involved in big decisions such as agricultural calendars and princely education, besides the Franciscans among the poor; the closed Japanese, besides trading in Nagasaki, still had selected people in Nagasaki watching and studying whatever stuff/tech the Dutch had; silver coins were often traded between china-philippines-mexico; The big decisions about opening/closing, or about tolerance/persecution of Christians were influenced by contacts with Europeans, including personal contacts of the daymo/shogun or imperial officials, dinners between the shogun and strategically drunk ship captains, and paranoid delusions about Spanish invasion from the Philipines or Christian help to the deposed chinese dynasty.

Moreover, even the concept of "closed country" does not make sense unless there is a "why?" or "against whom?" How can one discuss why they closed their countries with no account of what the europeans were doing there before/after the closure, or what people expected to happen if they opened up?

I think there are usually two possible meanings of the term.

1. In the same time frame as the European.Middle Ages, e.g. somewhere between 500 to 1500 AD

2. For an era that is deemed to be roughly similar to European feudalism.

See e.g. baike.baidu.com for 中古:

中古是指在西方文明史上意指约从公元500年到1500年之间一千年来的欧洲历史。

(rough translation: "Middle ages refers to the thousand years between 500 CE to 1500 CE in the history of Western civilization

and further down the same page:

①三古之一，较晚的古代，次于上古的时代。在我国历史分期上多指魏晋南北朝隋唐这个时期。 ②指封建社会时代。 ③指书体演变过程中的大篆。

(rough translation) 1. One of the three ages of old history, before later old history and after early old history. In our country it usually refers to Jin, Wei, Northern and Southern Dynasties, Sui and Tang [Wikipedia says that this time frame covers 266 CE to 907 CE] 2. the era of feudal society 3. (the term for a very ancient Chinese writing style)