[Vuejs]-I want to be able to set a "name" for a user in Firebase when using .createUserWithEmailAndPassword

You need to make an additional API call to set the displayName on the user:

firebase.auth()
    .createUserWithEmailAndPassword(this.email, this.password)
    .then(
      user => {
        return user.updateProfile({displayName: this.displayName});
      },
      error => {
        alert(error.message);
      }
    );