Well, where a tame parrot or house rabbit is concerned, those animals are often found by neighbors and taken to a shelter. The shelter scans all animals that come in and this is how they would find your contact info. One of my old friends once found an African Grey that just flew up to him and landed on his car. He decided to just keep it and not look. Under those circumstances, you really just won't find your bird. But if he had taken it to a vet or a shelter, the owner would have been found immediately (provided it had a microchip, of course).
You make a good point.