Yes, but that's as much his origin as Dick's was being from the circus. It didn't really impact on his actual nature after it was referenced in his origin. It was only really reintroduced for "A Death in the Family" to drive the plot.
Jason comes across as a pretty much perfectly normal kid otherwise, somewhat more short tempered than his previous incarnation or Dick, but NOT as some sort of "street tough".
And his loathing of Garzonas had little to do with his social standing, beyond the fact if gave him immunity. ANY Robin, and indeed most people, would have acted that way.