"Byrdes of on kynde and color flok and flye allwayes together."William Turner ~ The Rescuing of Romish Fox (1545)
You have a very interesting question. All animals with feathers are in the Class called birds or Aves and all the warm-blooded vertebrate animals with hair and females that produce milk are in the Class called Mammalia. Other Classes include Amphibians, Reptiles, and Pisces. Each animal within their class is then further defined by Orders, Families, Genera, and Species.
All dog breeds belong to the one species Canis lupus familiaris. The dog is descended from the wolf and the evolution of such widely differing breeds has been heavily influenced by conscious human selection, in addition to natural evolution. On the other hand there are about 10,000 species of birds. So the difference between the American Robin and the Black-capped Chick-a-dee is similar to that between sharks and goldfish, or dogs and cats, or lizards and snakes.
Birds have an innate ability to recognize their own species and look for mates based on their song, color of plumage, and behavior or distinctive courtship displays. However closely related bird species can interbreed, producing hybrids. One example is the Baltimore Oriole which can hybridize with the Bullock's Oriole where their ranges overlap in the Great Plains.
Birds in captivity can also hybridize like the Harlequin Macaws, Catalina Macaws, and many others. However birds prefer their own species in general. So, sorry, I don’t think we’ll see Stardinals anytime soon. But if you do, send me a picture.