This is an extremely interesting question, Biryani.
Just thinking aloud (relating to my own experiences): maybe it's due to the fear of causing disappointment by not living up to (possibly unreasonable) expectations? Not being able to live up to those expectations may have the possible consequences of being denied opportunities or being excluded from a community, or indicating shortcoming(s) in you and making you question if you were as good/special as you were told/had believed — that you were pretty "normal" after all. That would explain having the need to prove otherwise.
Although that would beg the question, why would this fear exist at all? The answer to that, I believe, may lie in the fact that we're fundamentally social beings, with a strong desire to belong. Not being able to fit into a group would've historically led to ostracization/abandonment, and thus possible death, so this fear may be deep-rooted in our brains as a survival mechanism to help us conform to the laws (both explicit and implicit) of a group.
This instinct may not be as useful in a modern society though, now that it's possible to survive independently due to an abundance of resources and access to basic necessities being dependent on entire supply chains instead of being controlled by any specific group that one is part of. This allows the luxury to attempt self-reflection, and to realise that even if you're not living up to someone's expectations, you have the option to seek the company of other individuals/communities where you will.
Or better yet, that maybe you're not obligated to conform to people's arbitrarily placed expectations and fit in at all — you, as an individual, have the first right in setting your own standards for yourself. This realisation, along with being more compassionate towards yourself, can lead you to a greater acceptance of your flaws and limitations.