Polyonymous comes to us from Greek. The "poly-" part means "many," and the "-onymous" part derives from the Greek word onoma or onyma, meaning "name" - so a reasonable translation of "polyonymous" is, in fact, "having many names." There are a number of other descendants of "onoma" or "onyma" in English, including "anonymous" ("having no name"), "pseudonym" ("false name"), "eponym" (someone who lends their name to something, or a word that comes from someone’s name), and "patronymic" (a name taken from one's father). Even "name" itself is derived from the same ancient word that gave rise to Greek onyma, making it a distant cousin of all these name-related words.
Word History
Etymology
Greek polyōnymos, from poly- + onoma, onyma name — more at name