Definition of elaphine: of, relating to, or resembling the red deer. From the Latin name of the Red deer Cervus elaphus From the Greek for deer, which is ἔλαφος (élaphos)
But why? The suggestion is that elephants have tusks, which are allied to horns, and at one time the language and classification of animals was loose and imprecise – so deer and elephants and cows and walruses and antelope would all be the same category.
That doesn’t really explain it because if that was the explanation then lost of animals besides deer would be called élaphos, which they are not. Elephants, for example, are Loxodonta – although they are in the family Elephantidae. I could elephants could have been reclassified at some later date, but then why stick poor old deer with the name of something that they are clearly not? And what about antelope, which have horns – they are bovidae. The Rowan antelope, for example, is Hippotragus equinus.