Premise 3 states that each homeowner keeps a different type of pet. By strict logic you can get all the way to the end and state for certain that the German does NOT keep dogs, cats, horses or birds, but it does NOT tell you he must therefore own a fish. The German could own a rabbit, or an emu for that matter. Unless you state that someone MUST own a fish the German could own anything (except what the others already own).
By asking who owns a fish you ASSUME that someone does and that someone is the German, but that's only an assumption - hence the need for another statement that someone owns a fish (some actually think this answer is the actual riddle and that only 2% of people would question whether anyone even owns a fish).