Which is the first movie to show having multiple personality disorder for one of its characters?
The novella in turn represents one of the first stories to reference or be inspired by then modern day psychological medical analysis of schizophrenia, split personality, disassociative identity disorder, or multiple personality disorders.
Specifically, inspired by one of the first patients to be diagnosed as such.
Louis Vivet, a mental patient who was suffering from split-personality disorder, caught [Author] Stevenson's inspiration while developing the story https://en.wikipedia.org/wiki/Louis_Vivet
The use of a physical transformation is just an Allegorical literary device used to represent the internal struggle of a man fighting himself for control. Jekyll, potion/catalyst aside, notices himself fall deeper and deeper out of control of himself, his descent into madness, like someone who sees MPD symptoms appearing.
The story is littered with references to MPD like symptoms.
With every day, and from both sides of my intelligence, the moral and the intellectual, I thus drew steadily nearer to the truth, by whose partial discovery I have been doomed to such a dreadful shipwreck: that man is not truly one, but truly two.
This story had been redone multiple times since the 1900s, first pre-film (aka theatre plays), then silent films, then talkies, color, 3D, modern film, etc.
The Three Faces of Eve 1957 is the earliest I have seen.
The main character, Eve, suffers from blackouts and is unable to recall where she has been during that time. Her doctor soon suspects that she has two different personalities. The whole film is based on her condition and recovery.
Arguably "M" (1931) fits.
Here is a monologue:
„Immer muß ich durch Straßen gehen, und immer spür ich, es ist einer hinter mir her. Das bin ich selber! (…) Manchmal ist mir, als ob ich selbst hinter mir herliefe! Ich will davon, vor mir selber davonlaufen, aber ich kann nicht! Kann mir nicht entkommen! (…) Wenn ich’s tue, dann weiß ich von nichts mehr… Dann stehe ich vor einem Plakat und lese, was ich getan habe, und lese. Das habe ich getan?“
"Always I have to roam the streets, and always I feel someone tailing me. That is myself! [...] Sometimes I feel like I'm running from myself. I want to run away from me but cannot. I can't escape myself [...] When I do it, I don't remember anything. Then I am standing in front of a billboard and read what I have done, and read. This was done by me?"
It's not really the full multiple personality plot device. However, it does show aspects of the medical disorder.