cognomen vs maiden name

cognomen and maiden name both are nouns.

Word	Noun	Adjective	Verb	Adverb
cognomen	Yes	No	No	No
maiden name	Yes	No	No	No

As nouns, maiden name is a hyponym of cognomen; that is, maiden name is a word with a more specific, narrower meaning than cognomen:

cognomen: the name used to identify the members of a family (as distinguished from each member's given name)
maiden name: a woman's surname before marriage

cognomen (noun)	maiden name (noun)
the name used to identify the members of a family (as distinguished from each member's given name)	a woman's surname before marriage
a familiar name for a person (often a shortened version of a person's given name)