Dunno when/how a game is classified a classic, but since PS2 is from the 6th gen, guess I have some suggestions! =D
Ultimate Ninja 5 is pretty cool, I think. Don’t like the anime much, but gameplay loop still feels pretty good nowadays. Only released on PAL and NTSC-J regions, though.
Dragon Ball - Budokai Tenkaichi 2 is also pretty fun, coming from someone that also doesn’t like the series it comes from.
I guess Godzilla: Unleashed could count as fighting game too? If so, I recommend it too.
Not a stats guy myself. At most, I get trophies out of something that feels a bit like OCD. Ultimately, I just want to have fun with my games, and not to boast them around. And tbh, one of the reasons I started dropping Steam was precisely the enforced social features (still use it, but now I’m finishing more games than I get instead).