Well well well, I knew you guys would debate about it...
There are indeed 3 possible answers, you all came up with them:
- We can't determine who keeps the fish, because the fifth pet can be any animal: the word "fish" isn't mentioned anywhere but the question. That fish is the fifth pet is only our assumption, so the first possible answer is that: we can't. (Stitchawl, Bilbo)
- If you do it the normal way, it's only a logical and technical process: you assume that fish is the fifth one, and then after some minutes you're sure that the German keeps fish. (Billi, HIMS)
(as for Paul, I'm not sure which side he's on

)
However, the full answer I was expecting has 2 parts: first, you conclude that the fifth pet doesn't have to be fish, and second, you use all the clues to exclude 4 people (Not Brit, not Dane, not Swede and not Norwegian).
In other words, you can't determine who keeps the fish, but you know who don't.
So... nobody's perfect, but together we can solve the famous quiz! It's rumored that Einstein said only 2% of the people in the world could solve his quiz, but we don't know if it was really Einstein's. And I think the world is also more intelligent now than it was in Einstein's time, because many people can come to one of 3 answers pretty quickly.
Over to... er, Stitchawl, or Billi, or HIMS. (Paul's and Bilbo's answers were not very clear...

)