The fight is between Islam and the elephant, with the donkey off playing around insisting there is not fight.
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".