Iirc it's even funnier: the relevant case law comes from Naruto v Slater. A case about a monkey taking a selfie and a photographer failing to acquire copyright of it (https://en.wikipedia.org/wiki/Monkey_selfie_copyright_dispute).
The copyright belonged to whoever shot the selfie, but because it was the monkey and animals aren't juristic entities, they can not hold copyright. Therefore, as it stands and as new case law outlines, AIs are compared to monkeys, in that the copyright would fall onto them but it's not a juristic entities either, and therefore copyright just vanishes and no one can claim it.
The wikipedia page suggests current cases on generative AI directly build on this.