One would argue that there is a Handesregistereintrag (record of commerce at the officials) that might help, but it only contain information about the seller including contact details and what the does, and not domains. And the record is not needed for small businesses.
TLDR: Germany seems to have hurdles for fraudsters, but they are easily taken by simply copying information from legit stores.
Adding the legit domains to the Handelsregister doesn’t seem like the worst idea to me. However, as the digital access to government services is still basically non-existent this would lead to a whole lot of additional bureaucracy and slowed down processes.
Let's say I set up a site that's critical of an authoritarian government. I fund it with sales of merch and books and such.
I want to be anonymous - for obvious reasons - but if I have to register my details I can't be.
Also, accountability doesn't work without international authority. Some countries are more enthusiastic about accountability and the rule of law than others, and the ones who aren't can make money by selling "credible" domains to bad actors.
Of course after a while those domains will become less credible. But there are a lot of TLDs out there now, which makes the system very difficult to police without international cooperation.
In reality I can run a scam operation from a beach in Thailand, bank the money, shut it down, then run a very similar scam operation from a beach in Vietnam or Costa Rica. That won't change until there's some kind of international cyberpolice agency which will hunt me down across borders.
But then you get the anonymity problem.
Not simple, and no registration system will fix this.