Bakugou is more obviously and blatantly a Gokudera situation, though his capacity to like, respect and submit to people straight up does not exist
Hibari doesn’t win the prove anything, he just think this is the natural order of things because he’s an entitled brat. At all times, he is calm, disinterested, and generally pretty bored. Having his absolute reality challenged by actually getting curbstomped just makes him irritable and eager to double-check, because obviously something went wrong the first time. He views “something to strive for” as “you let me become stronger and I’m coming for you” which is his version of comfort because he is absolutely fucking incredible as a character.
Bakugou is a narcissist who believes he’s the best because he wants to be the best, and when that absolute is challenged, he has a cataclysmic meltdown. Even the slightest hint of being considered ‘unworthy’ – Todoroki failing to overcome his trauma once he left the heat of the moment – he flips his goddamn lid. His insecurity is practically choking him to death. He doesn’t just want to be on top, he wants being on top to mean something, especially to people around him.
Hibari doesn’t really…care? About any of that? If you disrespect him he gets mad, if you beat him he gets mad, and while he is extremely reticent to actually admit he values his friends over his pride (or at least respects their pride as much as his own), he’s generally straightforward and doesn’t really care that he has a small army of loyal followers at his beck and call. He is a remarkably un-self-aware person.