Absolutely nothing I can read about this regulation, apart from angry internet comments, leads me to the conclusion that someone would have to register in every EU country; additionally, that's not how any other EU regulation works.

For example I don't have to register in every EU country if I make a social media site in one EU country. I only have to register it in my country and that agency may or may not forward the information to others. The law doesn't have to say this - it's implied.

My friend, this is how lots and lots of EU regulations work. See GDOR, the AI act and probably a bunch of others.

This is the kind of stuff that's in Draghis report, and it's economically equivalent to tariffs.