Akela isn’t a dog so much as the essence of dog. Just as man binds himself to a chosen totem, so is she wrapped in a form not of her choosing. She is the Hound because that’s how the world has defined her. It’s so overpowering, so forceful a definition that even in her own mind she embraces the term without thinking.
She is a Hound, but she isn’t a dog, and Jon’s only just beginning to understand the distinction.