Yes, it's normal and yes, she will grow out of it. Her other self is a kind of imaginary friend that may serve many purposes. Some such friends take the blame for misbehaviors, some are companions that take away loneliness, and others allow a child to experiment with different roles and identities. Creative children, firstborns, and girls are all more likely to invent imaginary friends.
Provided your child is otherwise socially appropriate, is involved in the world of play and other children, and is on target developmentally, rejoice. Show polite respect for this other self but don't go overboard and act as if it's too real. Your daughter knows it's all pretend. Do make her pay the consequences for misbehaviors that any one of her selves does.
Finally, don't try to argue this self away. The period of childhood when these imaginary companions join your family is all too short enjoy the magic.