This is kind of a cool question. Scholars typically treat adulthood as as the default and childhood as the "marked" class, so they look for children's culture and what defined defined a child.
You're asking about a giant swathe of the world. Unfortunately I can only talk about Latin Europe. But it's clear that western Christians absolutely had a concept of adulthood, although like today, there was no one, single, hard and fast boundary. Local and class variations in particular prevent us from saying something like "marriage" or "working outside your parents' household" as the one great adultmaker.
Medieval Latin Christendom actually inherited from ancient Rome the legal concept of adulthood by way of intellectual capacity! Medieval canon law, following Gratian, establishes 12 and 14 as the lower bound on marriage age for girls and boys respectively, with 7 set up as the lowest age for betrothal (the latter case was flaunted boldly by nobles, of course). The reason for the age boundary, in legal theory, was the definition of a valid marriage by consent, not consummation or parents agreement. The wife and husband's mutual consent to be married was the defining feature of what made them married. They had to be able to give consent, and kids couldn't do that! (Case 30, q.2; you have to scroll down a bit).
We see a similar development in monastic life in the later Middle Ages. Waves of monastic reform repeatedly try to ban child oblation, the practice of 'donating' a child to a monastery to establish spiritual and financial ties. Monastic life, the monks and nuns write, is a spiritually-grueling challenge that must be freely chosen by someone able to choose. (In practice, we know that nobles often earmarked specific children for later entry into a monastery anyway, and would have educated those children towards that end--including in convent schools. But the vows themselves needed to come later.)
Civil law likewise established boundaries. In England, propertied orphans became wards of the state, i.e. the crown or a proxy guardian controlled income from their properties until they reached their "perfect years of discretion and ripening of age." (parfight yeres of discrecion and rypenes of age, William Caxton, 1483).
Beyond law, we can point to more practical considerations. University students at the 'undergraduate' level (think ages 15-20 or so) could have been considered legal adults in some cases, but were typically still funded by their parents--we have numerous letters attesting to sending and receiving (or withholding!) money. In late medieval northwest Europe, girls and boys would often spend their teenage years into their early twenties working and sometimes living outside their home. This could be aristocratic young women living with another family to forge a broader social network with an eye towards marriage, or rural boys and girls moving to a town to work as domestic servants to earn a bigger dowry or more attractive financial base for a future household. This period of "life-cycle" work would cease with adulthood and marriage.
On the other hand, in a place like the Italian city-states, establishment of a new household would not have been a good marker for adulthood. When sons got married--often rather late, urban Italy is the place in Europe where we see the stereotypical "older guy, younger girl" marriage as a systemic pattern, even outside the highest class--they often moved their new wife right into their paternal household.
So overall--yes, medieval western Europe had a definite idea of adulthood, although it wasn't a hard and fast boundary.
There's always room for more discussion and exploration, but there have been some similar questions in the past that might have answers for you.