It’s not illegal for a person to learn things online. That’s one of the original purposes of the “world wide web” when it was opened to universities.

It is illegal to copy someone’s brand and use it to make money. These chat bots are literally charging people to take input like “write a story in this author’s style” and outputting a story that is a poor mimicry. The main problem is they are charging money based on someone else’s trademark. Not that they write a similar story.