You can, but it would be 'il est mon elephant.' (C'est = this is/it is)
If you go to the bottom of the "Êtres/Avoir" lesson on desktop, it explains how "ce/c'" is an idiomatic expression, like saying "that/it is..."
This is my elephant that only eats apples. Over there are the monkeys who also only eat apples. Oh, and thats my "It" who eats whales.
I'm really struggling to parse different words when hearing a new sentence. I spent forever trying to come up with "C'est mon, il est femme" which made no sense. When you're fluent, you're subconsciously anticipating logical conclusions to sentences and filtering things out based on context. Having to consciously do it is a frustrating mess.
This is the most useful phrase to learn in French. For me, at least. For others, it is probably irrelephant. Yes, I just went there.