If you want to actually “get” a domain name from icann, you are required to give them identifying information. There is no way around that. Except for the part at the end of this reply.

You can go through incognito or njalla and maybe even someone else, but at that point it’s “their” domain and you’re just renting it. The difference may matter for your purposes. Think critically about what you are trying to avoid and accomplish.

It’s not clear to me what you as a self identifying European expect to accomplish by not verifying your id with a group that asks for it in order to comply with the privacy law you imply you’d prefer to have the protections of. This is another area where I would encourage you to think critically about what you’re trying to accomplish and what material conditions surround you.

If you don’t want to do all that, just break the law. There’s a bigass database of ids from a breach of some company that does in person id scanning. Each one is a few bucks and you could easily pose as one of those people.