Although there aren't any "exclusive" Super Mario games on the PS2, there are a few titles that feature Mario characters or are related to the Mario franchise. One notable example is:

Ensure your console can run homebrew via FreeMcBoot.

: Some ISO versions include community-made high-definition texture packs and improved character models. Installation and Compatibility

The PS2, being a Sony console, has never had an official Super Mario game. The closest thing to a Mario game on the PS2 would be Mario & Luigi: Partners in Time, but this game was actually released on the Nintendo DS and Game Boy Advance, not the PS2.

What do you want to play on? (PC, Android phone, Steam Deck, or an old console?) Which specific Mario game are you trying to play?