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

https://www.hamburg.de/hundegesetz/