I don't have a solid answer, but I can talk about this topic in terms of fighting games since I'm somewhat knowledgeable in that category.
One factor is that fighting games more than most game genres are more intimate, meaning that it's just you and your opponent. Your skill does not depend on a race track or map design, there is very little luck, you don't have teammates, and overall fighting games depends heavily on your core skills. It's 1v1 so you have nothing to fall back on when it comes to excuses and due to the nature of fighting games, the skill gap is very noticeable. You may think you are good but then you can face someone new who absolutely stomps you and it becomes a reality check.
But for games in general, it basically comes down to human nature. We just do not like losing and being put down a peg by other people. We want to feel good by winning and pulling off the cool combos that we see top-level players perform. So like you said, it comes down to self-worth and being viewed by our peers as "cool" or "valuable".
