Rex vs adult male

Rex and adult male both are nouns.

Word	Noun	Adjective	Verb	Adverb
Rex	Yes	No	No	No
adult male	Yes	No	No	No

As nouns, adult male is a hypernym of Rex; that is, adult male is a word with a broader meaning than Rex:

Other hypernyms of Rex include man, crowned head, monarch, sovereign.

Rex (noun)	adult male (noun)
a male sovereign; ruler of a kingdom	an adult person who is male (as opposed to a woman)