It is rare for someone brought up the way Harry was to turn out the way he did. Of course we are influenced by our upbringing and the people we are surrounded by during our childhood. We are born with certain personalities, but they will show according to our parents/guardians.
Being bullied in our first years at school wil make us either insecure or stronger. Being treated as a servant and an inferior will mostly make us less confident.
The difference between Harry and his father is probably that Harry appreciates his friends and his new surroundings more because it all came unexpectedly. He had thought he'd be stuck with the Dursleys forever - childhood seems to last forever when you're 10.
I always admired the way Harry has been able to stand up to the Dursleys, it takes a great and strong personality to do so.
As a fiction character this also put hi in much the same position as Voldemort, Snape and Sirius, some of the most interesting characters in the books. Oddly enough not Dumbledore who grew up with loving parents and a "normal" family life - at leat until he was about 11 (?)