It didn't really impact on his actual nature after it was referenced in his origin.
This is like saying that Cass looking Asian doesn't have an influence on how she gets written. It wasn't hammered into us every time he appeared that he wasn't from the same privileged background as Bruce, but that doesn't stop the way Starlin characterized him from following a lot of stereotypes associated with lower class. And Starlin's characterization being responsible for both a Death in the Family and the way Jason was remembered after his death, I put more importance on it than on others.
And his loathing of Garzonas had little to do with his social standing, beyond the fact if gave him immunity.
Of course, that's the point. If you're rich, you can get away with anything, that's how it looks. You can be Garzonas or you can be Batman. Jason noticed that.