Earlier quoted context omitted.
In Germany we have an approach with a somewhat similar effect. For any site with an commercial intent (which is pretty loosely defined) it is mandatory to have an Imprint with the person representing the company, the address of the HQ as well as the companies registration number and court location. It makes it somewhat more transparent what company is behind the site and gives you information you can lookup in public…
My rude opinion: imprint is nonsense. It does not protect you from fake shops at all. It’s no brainer to copy one from another shop. As a legit seller you can be sued by shady layers for errors in imprint. Who is looking in public registers while shopping online…
I don’t think it’s completely useless, but it’s certainly not perfect either. As the parent comment suggested having a business register It’s legit domains would probably makes sense from a consumer protection point of view.
However, with the current state of the digital administration in Germany this change would introduce so much overhead that it would lead to a lot of justified opposition.