I have also heard of people insisting that birds are birds, not animals, and I believe I know where the confusion lies. It comes from a fuzzy memory of what we learned in elementary school. In 4th or 5th grade, we are taught the different classifications of living creatures (plant/animal, vertebrate,invertebrate, etc.)
What these people really mean, but are remembering incorrectly, is that birds are not MAMMALS. Birds are in a classification seperate from mammals, reptiles, amphibians and insects. This is a difficult concept for 10 year olds to grasp because they think birds are warm blooded and should be called mammals. Actually, birds are more closely related to lizards, but it is tough to convince a kid that because birds are cute and lizards are icky. Somehow, in the attempt to impress upon young kids the seperation of birds from other creatures, they confuse it to mean that birds are not animals. They certainly are, though a different type of animal than all others.