That said, you cannot be a mother to your child. Your roles now are approximately equal, but they will inevitably change over time. Feeding and changing diapers is enough for now, and that indeed can be split between partners, because baby cannot really tell the difference right now.
For further normal development, babies need to develop a bond with their mother. This gives them a fundamental sense of security. It is one of human baby's primary needs which is also observed in most mammals.
Children later become attached to their fathers. It is a different experience for them, which involves more "rational" thought instead of primary urge to bond with their mothers.