Literally anyone else could easily generate the exact same image, and are they going to be in violation of my copyright now?

It is already the case that if an AI generates an image that happens to be effectively identical to a copyrighted one the person who generated the image can be in violation of that copyright. It doesn't matter how the copyright originated.

In the case of your cat example, though, the solution is trivial. Use a random seed. There's far too many potential images any given model could generate to ever copyright all of them, or even a tiny sliver of them, and if that did miraculously happen just train the model a little more and you get a whole new set of outputs. It's unfeasible.