paper bag vs doggy bag

paper bag and doggy bag both are nouns.

Word	Noun	Adjective	Verb	Adverb
paper bag	Yes	No	No	No
doggy bag	Yes	No	No	No

As nouns, doggy bag is a hyponym of paper bag; that is, doggy bag is a word with a more specific, narrower meaning than paper bag:

paper bag: a bag made of paper or plastic for holding customer's purchases
doggy bag: a bag for food that a customer did not eat at a restaurant; the transparent pretense is that the food is taken home to feed the customer's dog

Other hyponyms of paper bag include doggie bag, grocery bag.

paper bag (noun)	doggy bag (noun)
a bag made of paper or plastic for holding customer's purchases	a bag for food that a customer did not eat at a restaurant; the transparent pretense is that the food is taken home to feed the customer's dog