dancing partner vs dancer

dancing partner and dancer both are nouns.

Word	Noun	Adjective	Verb	Adverb
dancing partner	Yes	No	No	No
dancer	Yes	No	No	No

As nouns, dancer is a hypernym of dancing partner; that is, dancer is a word with a broader meaning than dancing partner:

dancing partner: one of a pair of people who dance together
dancer: a person who participates in a social gathering arranged for dancing (as a ball)

Other hypernyms of dancing partner include collaborator, cooperator, pardner, partner, social dancer.

dancing partner (noun)	dancer (noun)
one of a pair of people who dance together	a performer who dances professionally
	a person who participates in a social gathering arranged for dancing (as a ball)