Shadow of the Colossus was the first game that gave me feels that I couldn't articulate. It re-framed my idea of what a game could be, and gently pushed me to think about games as a medium.
I was too young to play it when it came out, but watching my dad play it was the most mind blowing things to me as a kid. I was completely immersed in it then, and its still my favorite game of all time.
Dragon Ball Z Budokai Tenkaichi 3, this game is perfect as a Dragon Ball Z simulator, even for today's standards, I wanted to mention an exclusive title, but many of the PS2 games have already been ported to other platforms, which speaks very well about the PS2 itself.
It's my favorite PS2 title because it completely caught me off guard. This was my first music/rhythm focused game and it got me hooked on the genre in the following years (Harmonix made Amplitude a few years before the Guitar Hero wave started).