domestic dog vs doggy

domestic dog and doggy both are nouns.

Word	Noun	Adjective	Verb	Adverb
domestic dog	Yes	No	No	No
doggy	Yes	No	No	No

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

domestic dog: a member of the genus Canis (probably descended from the common wolf) that has been domesticated by man since prehistoric times; occurs in many breeds
doggy: informal terms for dogs

domestic dog (noun)	doggy (noun)
a member of the genus Canis (probably descended from the common wolf) that has been domesticated by man since prehistoric times; occurs in many breeds	informal terms for dogs