I agree with John. We usually don't view ourselves as a title like a doctor, lawyer, etc. These are our jobs, not who we are. So we use "What do you want to be when you grow up?" not "Who do you want to be when you grow up?"
When we say "Who do you want to be?" it refers to the *type of person* we want to be or *the qualities* we want to have. "I want to be a kind person, a wise person, a generous person..."