odd man out vs doer

odd man out and doer both are nouns.

Word	Noun	Adjective	Verb	Adverb
odd man out	Yes	No	No	No
doer	Yes	No	No	No

As nouns, doer is a hypernym of odd man out; that is, doer is a word with a broader meaning than odd man out:

odd man out: someone regarded as eccentric or crazy and standing out from a group
doer: a person who acts and gets things done

Other hypernyms of odd man out include actor, worker.

odd man out (noun)	doer (noun)
someone regarded as eccentric or crazy and standing out from a group	a person who acts and gets things done