Seems to me like producing an artist's stylised signature would be a trademark infringement.
OpenAI give lip service to the idea of not producing others' intellectual property - go ask it to explicitly make a picture of the genie from Aladdin.
It is certainly not trademark infringement, I can say that for sure -- no reasonable person would prompt ChatGPT to create a "new yorker style cartoon" and then think the result is produced by the actual cartoonist in mere seconds for their viewing pleasure. So there's no confusion in the marketplace.
The user could cause confusion in the marketplace of course, but that would be her doing, not the app's. Surely we can all agree that suing adobe illustrator for facilitating trademark infringement of logomarks and such would be silly?
It could be copyright infringement, which should drive home how absurd copyright is as a concept. Everyone's all up in arms about Anthropic reporting a user to the police today -- imagine if the thing they were reporting was that she had written a sacred symbol in her personal notebook...
https://openai.com/index/disney-sora-agreement/