His personality trait is pretty similar to any other overly cocky characters who happen to also be genius and thus believe their cockiness is justified. I won't say it's that edgy, generic maybe.
But that's the first impression.
Upon a closer inspection, his cockiness is perhaps just a facade put up merely to deal with Rhaast, given they can vaguely sense each other's emotions due to their connection; and the last thing you wanna do when you're against your literal mortal enemy is to show them your back (as evident by Rhaast occasionally taunts Kayn by claiming that they feel there is doubt in the latter's heart). And this is why they banter like salty good friends, when in fact they're just plotting to kill each other when the time is right. They seem like a duo that rely on each other to complement each other's characters, and it really does seem they complement each other just fine; but the reality is that one has to die in the end, and they both can't wait to get rid of the other and absorb their things as their own. If you think about it this way, Kayn is suddenly not such a shallow character.
Honestly, I really don't believe Kayn is as badly written as some would say tbh. In fact I'd say he's pretty solid. He has that generic flair going for him, the long black hair and young face, the long and big scythe, the shadow demon power, and cockiness and that stuff with being a genius, all of them recipes for the most generic DeviantArt OC; but then manages to distinguish himself by actually having a meaningful character just fine, and act in such a way he's a self conscious parody of the supposed edginess with underlining layers of his own rather than just being defined as one. I'd like to think he's okay.