In terms of multiple personalities, it is Dissociative Identity Disorder (DID). Most of the patients are caused by childhood trauma. Due to psychological self-protection, they will deliberately isolate embarrassing past events and memories. Or put this part of the memory in another personality in some form.
In the play, David once said that he wanted his daughter to have a better childhood, which implied that his childhood was not very good.
And when Emily drew herself at the end of the credits, she drew two heads. It's because Emily has been through all this horrific, she's showing signs of multiple personalities herself, and she really has imaginary friends. And because of her father's experience, she realized that the so-called friend was just another personality of herself. So she drew two heads, not two people.
Another point is that David is not actually a psychiatrist, and everything about psychiatrists is based on childhood trauma. There is a scene in the play when David opens the diary that he thought was full of Emily's illness, but finds it is blank, and the headphones are still in the box; there is another scene where David was originally writing in the notebook, and then the camera changes to David wearing headphones. Awakened from a dream. These all suggest that David was in fact just asleep, and the memory associated with his personality led him to believe he was taking notes.
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