Ps2 had a great selection of games, it's the one Sony console I'm accustomed too. One of my favorite games in it is definitely Shadow of the Colossus but there's already a thread on that, if anyone is interested in my thoughts on it I'll leave a link on it below.
https://warioforums.com/index.php?threads/the-ico-series-shadow-of-the-colossus-last-guardian.2326/
Aside from that having fond memories of playing SF Alpha collection as well as Capcom fighting jam with friends after my Karate classes have been the best. Fighting games have been a staple for Sony and the controller is comfortable enough to use although many would just go against it for an arcade stick which is also available for consoles in general. First time getting a PS2 had about 3 main games that I started with two of the games being really good Klonoa 2 and ZOE. Zone of the Enders was a Konami game that Hideo Kojima(who is popular for working with the MGS game series) also worked on, it was a fun mech game and the designs for each giant robot is lovely especially the main robot you control namely Jehuty. You basically go around various mission's taking on opposing mech's bent on doing whatever they want i.e. destroying habitats and settlements. Don't remember much of the game except for the third boss namely being quite sad after you beat it since it only wanted love and out of desperation started destroying things, the gameplay and even style reminds me heavily of Gundam Zeta(anime).
The third game I absolutely hated it was called Fur Fighters and it sucked ass. The controls are wonky, you play as ugly furries trying to save other ugly furries. That aside it just didn't seem very fun trying to use guns in a platforming game is really kind of dumb which is what Shadow the hedgehog suffered years later with the same sort of thing.