To play on your analogy I would go as far as to say that the donkey is so rabidly anti-elephant that he sees radical Islam as the "enemy of my enemy" and thus a friend. Blinded by his hate for the elephant he overlooks the true nature of radical Islam that he'd would happily open the door for it, not realizing that in doing so he'd secure his own demise. Radical Islam tolerates and even pretends to have common cause with the donkey because for the time being he is "useful".
Sort of. Part of the reason the donkey appears so rabidly anti-elephant is it is a craven coward. If the elephant were really as blood thirsty and evil as Islam, the donkey would be kissing the elephants behind...while pretending not to.
But deep down, the donkey really just wants to have fun, be in charge, do whatever it feels like, and not have any responsibility. It only hates the elephant because the world the elephant lives in is real, and the donkey feels this makes any ills that happen in the real world the elephant's fault.