A person with two or more well developed identities has the disorder called

I believe the answer is: dissociative identity disorder.

In dissociative identity disorder, there would be two or more distinct identities that would constantly tried to overtook the consciousness/existence of the patient.

This disorder is considered as a 'severe' condition and it is very common for the victim to experience time skip and not even remembering the things that they have done even when they're awake