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I'm guessing it's a cognate with the Spanish word for rabbit (conejo)?

From Latin via Anglo-French and Old French, but the Spanish also derives from the Latin. https://www.etymonline.com/word/coney

more likely from old french "conin"

"Conin" derives from "coniculus", which is what "conejo" derives from, so that makes all three words cognates, since they share a common origin.