In Germany it is straight up illegal. The UK law has provisions where a services has to name a responsible person or report specific things to the government. Obviously those can not be accomplished anonymously. In any case hosting a website anonymously doesn't work if you want to work within the law, any lawsuit against you will identify you.
> I don't even have to provide my personal info to register a domain name, I can run it on an IP-address
Which is totally irrelevant. I can also go into a store and take something without paying. The question is whether that is legal or not and what you need to do to keep it legal.