jkpate, that's an interesting point. However, wouldn't that mean that we'd at least be able to
understand cat? Maybe we would inherently speak it in a way that is too complicated for other animals to understand, but a simpler system would be within what we would understand, as part of understanding the more complex system. I think.
To say that humans are superior because we ended up with one trait over another is, well, childish. Oh, and dare I say, circular.
I'd agree with you there. I've never understood the need to make and adjust such claims in light of how they're frequently falsified either

But humans are the only ones that intentionally communicate with a learned system of symbols that have been agreed upon by a community.
What's unique to humans is the arbitrariness of the sign? I don't think that's true. The details are not something I'm particularly familiar with, but I think some other species use arbitrary signs. For example, apes have been taught to use bits and pieces of (arbitrary) sign languages, and prairie dogs have different dialects in different locations.
To phrase the question another way, then, why can't humans speak cat?
All species have special things that other species don't do - that is why they have their own niche and adaptive pressures resulted in them speciating in the first place. Humans, like other animals, have unique traits.
But then why do we assume that there is some special trait that is inherently better than the traits of all other species that allows us to use Language?
The impression I get from Chomskian approaches to language evolution is that there was a time when humans communicated like other animals then there was a genetic mutation (he says Merge) that allowed humans to have Language. And that missing linguistic link is all that separates us from animals and all that critically supports our ability to speak. Its presence and development is deduced from the "fact" that we clearly have a more evolved communication system, and so forth.
I'm question those assumptions.
If all species (or some?) just communicate "differently", then there's no reason to assume a basic logical/mathematical difference between the systems; rather, they may differ in complex ways, not just one better system replacing an inferior one.