Hamburg is not a U.S. state—it is a city-state (Bundesland) in northern Germany. Therefore, it does not have a "state bird" in the American sense.
Germany's national bird is the White-tailed Eagle (Seeadler). Hamburg's coat of arms features a castle with three towers (symbolizing its medieval fortress origins), not a bird.
If you're thinking of a U.S. state bird, you might be confusing Hamburg with a state like Hawaii (Nene) or New Hampshire (Purple Finch).