I'm planning a week-long stay in Hamburg next year. I want to come with my dog. I know the rules with pet passport, rabies vaccination etc, but I also found something about a Hamburg Dog Registry and couldn't really tell the registration and insurance rules also apply to tourists traveling with dogs?
It only applies to Hamburg residents. Note that you are still required to observe all other Hamburg regulations concerning dogs; mainly: keep your dog on a leash outside of designated dog zones (it's possible and common to get an exemption, but it requires you to pass a test and pay a fee - not something that's worth it or feasible for only a week), and pick up after your dog.
I don't think that part is relevant to you as a tourist, other parts of the law might be though. Details can be found here in case someone else wants to elaborate or you want to translate the text yourself