[pavon]

They are claiming both those things - copyright infringement and a trade dress infringement under the Lanham Act.

That said, their trade dress claim doesn't go so far to claim ownership of an entire style, it is the use of that style in association with their names that is the problem. For example "draw a stick figure cartoon dog" is fine but "draw a dog in the style of xkcd" is not, by their reasoning. And you certainly can't advertise that the model can make images in the style of these artists in ways that might be interpreted as the artists being involved with the company.