The Chinese Calendar

The Chinese calendar year is the same length on average as the Gregorian year, because they have leap months.

The Chinese calendar follows a 60 year cycle of when leap monthd are inserted into the year.

A lunar month is 29 or 30 days long (because it's exactly 29.53 days from one new moon to the next) so a lunar 12 month year is just over 354 days long, which is almost 11 days short of the tropical year. To keep the calendar in sync with the solar year, every couple of years thirteenth month must be inserted into the year.

The year 2020 (4717 in the Chinese calendar) had a leap month between it's 4th and 5th regular month, and this year (4720) there was a leap month between the 2nd and 3rd month.

Generally, there are 22 leap months in a 60 year cycle.

This calendar would have an error of one month every 600 years

The Jewish calendar

The Hebrews have a similar system, inserting 7 leap months in a 19-year cycle (metonic cycle)

This one is even more accurate, drifting away one month only after more than 6000 years.

So these two calendars both have an average year length of 365.25 days, so your age in both of them is about the same as in the Gregorian calendar.

The Islamic Calendar

What you are looking for is the islamic calendar. It is purely lunar and doesn't have leap months. Every year is between 354 and 355 days long, meaning that after 33 years in the gregorian calendar, 34 Islamic years have passed. So no, this wouldn't make that much of a difference with the drinking age in the US, but you could get away with alcohol 8 months earlier that way. Of course, ignoring the fact that most Muslims that unironically use the islamic calendar for everything also don't drink