The German/Prince combination only has one option according to the chart: House #4, which leaves the other the other remaining clues with only one option.
Pall Mall/Birds combination
If we start with the German/Prince combo (since it's the only combination that only has one option), all of the remaining clues will have only one option left!
The German/Prince combo is House #4, since the owner of House #5 smokes Bluemaster and the Brit lives in House #3.
The Pall Mall/Bird combination goes with House #3, since house #5 is the only other available option and the owner of that house smokes Bluemaster.
There's one house left where the Swede and his Dogs can both go, so we can put them both in House #5.
Now that we've filled in all of this information, the only spot left is the pet for House #4, which means that the German owns the Fish!