I've worked with animals enough to know that they definitely do have individual personality traits--some are shy, some are aggressive, some have greater or lesser energy, etc. Even siblings from the same litter can show remarkable differences. They also do have working brains and emotions, they take in information, they make decisions, and they even come up with new ideas (rats especially--they're clever lil things).
So it wouldn't surprise me if animals did indeed utilize cognitive functions similar to what humans have. But, beyond the E/I split that's already been noted, I'm not really sure how we'd assess what they are--I worked with my mother's 3 dogs for several years and still couldn't tell you what MBTI types I think they have. I could give you a vague approximation of their enneatypes (one is almost stereotypically sx-6w7ish) but beyond that, not much.
Others have pointed out that humans have developed the MBTI largely to suit human needs, so that's a further barrier to actually knowing in what way other species might think. What would Fe mean to a non-social species, for instance? Would being Ti-dom have any evolutionary purpose for, say, a coyote? What would be the cognitive differences between reptilian brains versus mammalian brains? Etc.
I don't have the answers here, but it seems premature to rule that other animals (remember, humans are also animals) don't have some sort of cognitive processes, though they may not work for them the same as they do in humans. People are only starting to study the brains of other animals, and MBTI isn't even "scientific", so it's unlikely we'll ever get a definitive answer on that.
(As you might guess, I personally think there's something to it, but that's mainly just my personal feeling on it.)