미국 나이 means "the age reckoning of western countries" or ""man-nai" (만나이) "
Quoted from https://en.wikipedia.org/wiki/East_Asian_age_reckoning#Korean
Koreans generally refer to their age in units called sal (살), using Korean numerals in ordinal form. Thus, a person is one sal ("han sal", 한살) during the first calendar year of life, and ten sal during the tenth calendar year.
The 100th-day anniversary of a baby is called baegil (백일), which literally means "a hundred days" in Korean, and is given a special celebration, marking the survival of what was once a period of high infant mortality. The first anniversary of birth named dol (돌) is likewise celebrated, and given even greater significance. Koreans celebrate their birthdays, even though every Korean gains one 'sal' on New Year's Day. Because the first year comes at birth and the second on the first day of the New Year, children born, for example, on December 29 will reach two years of age on the New Year's Day, when they are only days old in western reckoning. Hence, everyone born on the same calendar year effectively has the same age and can easily be calculated by the formula: Age = Current Year - Birth Year + 1
In modern Korea the traditional system is most often used. The international age system is referred to as "man-nai" (만나이) in which "man" (만) means "full" or "actual", and "nai" (나이) meaning "age". For example, man yeol sal means "full ten years", or "ten years old" in English. The Korean word dol means "years elapsed", identical to the English "years old", but is only used to refer to the first few birthdays. Cheotdol or simply dol refers to the first Western-equivalent birthday, dudol refers to the second, and so on.
For official government uses, documents, and legal procedures, a chronological age system is used akin to the system used in Western countries. Regulations regarding age limits on beginning school, as well as the age of consent, are all based on a chronological system (man-nai).