I really didn't "grow up" with the PS1, but I did play with the PS2 for a while. Here are some PS2 games I would recommend:
RPGs (non-Final Fantasy RPGs, because by now, you are probably getting tired of hearing that series being mentioned over and over again):
Dark Cloud (There's a Dark Cloud 2, which is called Dark Chronicles in other countries, but I have't played it yet)
Rogue Galaxy (The people who made Dark Cloud, known as Level-5, also made this game and Ni no Kuni)
The first three Jak and Daxter games (you can also get these games as a collection disc for the PS3)
Maybe Tekken 4, for boredom's sake.
RPGs (non-Final Fantasy RPGs, because by now, you are probably getting tired of hearing that series being mentioned over and over again):
Dark Cloud (There's a Dark Cloud 2, which is called Dark Chronicles in other countries, but I have't played it yet)
Rogue Galaxy (The people who made Dark Cloud, known as Level-5, also made this game and Ni no Kuni)
The first three Jak and Daxter games (you can also get these games as a collection disc for the PS3)
Maybe Tekken 4, for boredom's sake.