The short answer to this question is no, eggs are not considered a dairy product. A dairy product is milk or a food which is made from milk, such as butter, cream, or cheese. Eggs are an animal product that comes from avians such as chicken. So eggs are not considered a dairy product. A great way to remember this is to reference the food pyramid. In the food pyramid, eggs are in a separate category from milk products. Eggs and dairy have entirely different physical and chemical compositions.
Eggs and dairy are often linked together in the eyes of the public, for a variety of reasons. Both come from animals, typically coming from farms. In communities where dairy delivery is available, eggs can typically be ordered as well. Eggs and dairy are also often shelved near each other in the supermarket which may encourage consumers to categorize them together.
A dairy product comes from mammals like cows, goats, sheep, horses, yaks, llamas, camels, or other animals which produce milk. The precise chemical composition of a dairy item varies, depending on the animal it comes from. This is why some people who have trouble digesting cow's milk may be able to drink less rich milks, like sheep or goat milks.
Eggs come from avians like chickens, ducks, and geese, among others. With the exception of the monotremes, such as the duck-billed platypus, an animal which lays eggs cannot produce milk. Milk and eggs have a markedly different composition. This is why being allergic to one of these foods does not automatically make one allergic to the other.
The link between eggs and dairy may also be related to the fact that the two are often used together in recipes, and to the fact that strict vegetarians avoid both, out of concern for animal welfare. However, dairy products exclusively refer to products made from milk and should not be classified with eggs. Eggs may also be classified differently from other animal products by vegetarians since an animal is not slaughtered for this food item.