Yes, it makes sense, if your pet’s food is mamash chometz. You give or sell the pet *and* the food to a goy, and now you are doing him a favor, feeding his food to his animal. Since the food is presumably not anything you would ever dream of popping into your mouth, no matter how absent-minded you are, it’s not a problem to store it on your property and use it in this way.
But if the food is not mamesh chometz but only a mixture of chometz, i.e. if you look at the ingredients you find that wheat is down the list, and therefore is less than 50% of the total, then you don’t need to go through any of this. You can keep the animal and the food, and feed it as normal. As far as I can tell, this includes ALL BRANDS OF DRY CAT FOOD. I have never seen a brand of dry cat food where grain is a majority of the mixture. Even if there is any grain it’s way down the ingredient list. So there’s no shayla, it’s completely muttar to possess and use it. Of course it’s forbidden to eat it, not only on Pesach but all year, but you would never do that.