age vs drinking age

age and drinking age both are nouns.

Word	Noun	Adjective	Verb	Adverb
age	Yes	No	Yes	No
drinking age	Yes	No	No	No

As nouns, drinking age is a hyponym of age; that is, drinking age is a word with a more specific, narrower meaning than age:

age: a time of life (usually defined in years) at which some particular qualification or power arises
drinking age: the age at which is legal for a person to buy alcoholic beverages

Other hyponyms of age include age of consent, legal age, majority, minority, nonage, voting age.

age (noun)	drinking age (noun)
how long something has existed	the age at which is legal for a person to buy alcoholic beverages
a time of life (usually defined in years) at which some particular qualification or power arises
a late time of life
an era of history having some distinctive feature
a prolonged period of time