red deer vs American elk

red deer and American elk both are nouns.

Word	Noun	Adjective	Verb	Adverb
red deer	Yes	No	No	No
American elk	Yes	No	No	No

As nouns, red deer and American elk are synonyms defined as:

Other synonyms of red deer include Cervus elaphus, elk, wapiti.

red deer (noun)	American elk (noun)
common deer of temperate Europe and Asia	large North American deer with large much-branched antlers in the male
	common deer of temperate Europe and Asia