I've heard that grad students go through something very similar when they go to grad school: feelings of inferiority and fears of being "exposed" as fake when surrounded by so many other intelligent people.

I've never worked for a start-up but imagine the start-up scene works the same given the common stereotypes: success stories of 20-year-olds, hyper intelligent ivy-league drop-outs, etc. I don't know whether they're the rule or the exception to it.