These are different things to me. Say you feed your dog under the dinner table. Then the food you give is the dog's food, because it's the food that belongs to the dog. However it is not dog food (well, unless you're having dog food for dinner).
By the way, 'this is animals' food' would translate to 'to jest jedzenie zwierząt'
But it's "na użytek", not "na studentów" :)
In fact, this "na" in your sentence about animals sounds to me more like an anti-animal thing. For example "spray na komary" is a spray that kills mosquitos. "packa na muchy" is a fly swatter. "strzelba na słonie" is, well, a shotgun used against elephants :/