I apologize for taking my time responding to this. I wanted to convey my thoughts on the matter properly, so here we go.
Let’s start with the big one:
Identity: Markedly impoverished, poorly developed, or unstable self-image, often associated with excessive self-criticism; chronic feelings of emptiness; dissociative states under stress.
We all know who Kaiba is, right? Except… We don’t, really. We know he had a traumatic childhood. We know he is a rich CEO. We know of his accomplishments in Duel Monsters. It is almost impossible to divorce him of these aspects. We know that Tea likes dancing and Yugi likes puzzles. What are things Kaiba enjoys? At best we could say he likes dragons.
And you can’t say Duel Monsters. It is a painstaking labor for him. It keeps him up for hours doing research into the night. Not as a hobby, but through compulsion. Though there might be momentary satisfaction to gain from it, I’d be hard-pressed to say it’s for fun. During their second duel he even tells Joey “I don’t have fun.” He struggles with the concept as nothing is ever fulfilling.
You take Joey away from the card game and you still have a funloving, loud-mouthed, knucklehead …But what is Kaiba? He’s probably unsure of the answer himself.
“Dissociative states under stress”
He actually calls himself “nothing.” He is his own worst critic. He determines his worth by how well he can play Duel Monsters because that’s his thing.™ It’s my whole identity, my whole life is based around this so if I can’t be good at it then what’s the point.
“Suicidal threats, self-destructive or risk taking behaviors.”
This speaks for itself. Also, [spoilers] in DSOD Kaiba sacrifices himself so that Yugi may call forth the Pharaoh and then fades, for all we know, from existence forever without so much a goodbye, and consistently rushes headlong into danger for Mokuba. Almost as though he doesn’t care much for his own wellbeing or realizes he’ll be missed at all.
“Feeling fearful, apprehensive, or threatened by uncertainty; fears of falling apart or losing control.”
Do I even need to mention how many times the duel with Ishizu had him freeze up in abject horror? He was utterly shaken. Not only if he lost would he lose humiliatingly in his own tournament, but it’d be a grander reminder he has no control over his destiny after all and that absolutely terrifies him.
Empathy: Compromised ability to recognize the feelings and needs of others, or hypersensitivity.
Remember that time Joey literally died and two other people got sent to the infirmary which Kaiba casually disregarded for the sake of completing his tournament? But, Pegasus summons a toon Blue-Eyes and suddenly he’s highly upset like it burned all his crops, specifically, to slight him?
While that may not seem like a big deal to us, he actually says “you know how much I hate that thing!” In such a frustrated and hurt tone that it’s clearly very offsetting to him. He’s demonstrably apathetic yet pretty sensitive. Certain things seem to trigger explosive fits.
“Pattern of unstable and instense relationships characterized by alternating between extremes of idealization and devaluation.”
He’s always touting Yugi’s skills as a duelist and 100% sure Yugi can beat whoever he’s up against! Except for when he isn’t. Yugi’s losing? Disgraceful. Pathetic. I misjud– oh, Yugi’s winning? I knew it!! Yugi is the best!
This black and white thought process is often referred to as “splitting.” It can occur with anyone, (or anything) but in relation to an FP (favorite person); someone you’ve imprinted on and have a large attachment to, it’s the most drastic.
These can be people you’re so consumed with you don’t bother acknowledging anyone else which, Kaiba has a tendency of doing.
“Inappropriate, intense anger or difficulty controlling anger.”
Recall this outburst from season 4. He’s visibly distressed. And notice just how freaked out everyone is by this reaction. This isn’t usual Kaiba anger.™ He genuinely feels betrayed. He’s so disconsolate he doesn’t even give Yugi a chance to explain himself. If Yugi had unknowingly become his FP, this extreme reaction would make sense.
Hostility: Antagonism. Persistent or frequent angry feelings; irritability in response to minor slightly and insults.
I don’t even have to explain this one. [Insert any given interaction between him and Joey or the gang ever.]
“Frantic efforts to avoid real or imagined abandonment.”
*GESTURES WILDLY AT THE ENTIRETY OF DSOD*
Anyway, I don’t really know how to wrap all this up but, it’s worth noting somewhere that people with BPD struggle with interpersonal relationships and making friends as they have issues with emotional permanence. Meaning, they’re constantly needing to “prove” a feeling, or person is really there. They have trouble believing anyone will stay or are really there for them and tend to isolate.
This could explain Kaiba’s standoffish attitude and why he constantly wants to challenge Yugi to prove he is the best. He may also be seeking validation from Yugi that he is a good duelist. Because there is only now to him. The present defines him. “If I’m not the best at this very moment,™ was I ever the best?”
People with BPD constantly feel that they have to earn the right to exist, and constantly wage battle with themselves to be their version of what’s useful or the “best” so they can feel they’ve made their steak. Kaiba’s narcissism is a defense mechanism. He seems to fluctuate between overzealous manic episodes, and crushing, despairing, depressive ones.
At his core, he is insecure and attempts to fill the void with winning tournaments, or building an amusement park– any project or interest to keep him busy. However, he tends to hyperfixate to the point of obsession. All of which are symptomatic of Borderline Personality Disorder.