cleaning man vs cleaner

cleaning man and cleaner both are nouns.

Word	Noun	Adjective	Verb	Adverb
cleaning man	Yes	No	No	No
cleaner	Yes	No	No	No

As nouns, cleaner is a hypernym of cleaning man; that is, cleaner is a word with a broader meaning than cleaning man:

Other hypernyms of cleaning man include adult male, man.

cleaning man (noun)	cleaner (noun)
a human female employed to do housework	a preparation used in cleaning something
	someone whose occupation is cleaning
	the operator of dry-cleaning establishment