In some places in the world, such as the U.K., Indonesia, Cuba and Vietnam, 16 years is the age of legal adulthood and confers the right to enter contracts, marry and often, vote. Many more countries offer voting rights to legal minors, including Brazil, Argentina, Austria and Greece. If anything, the fact that we in the U.S. see 18-year-olds as suddenly mature is likely as much a product of the fact that we let them vote, rather than a cause.
The focus on 18 is an example of what I, a sociologist who studies the cultural meanings of age, call chronological essentialism; that is, when we believe that the number of years since a person was born is the most accurate depiction of how mature they are. This belief is dominant in our culture, but it is just that: a belief.
Chronological age only seems universal and objective because Western societies have agreed to measure time in a certain way and use that to keep track of their citizens. It's only in the last century that most governments around the world have attempted to collect reliable and universal birth records, and therefore have documentation of people's chronological ages. In the past, withholding knowledge of birthdays was a way that our country denied some people, such as those enslaved, personhood. As Frederick Douglass described, "I have no accurate knowledge of my age. ... [My master] deemed all such inquiries on the part of a slave improper and impertinent, and evidence of a restless spirit."