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:
Unlike emulation, native ports are rewritten to run directly on the PS2 hardware, offering smoother performance. Super Mario 64 super mario ps2 iso exclusive
There is no "Super Mario PS2 Exclusive." There is no hidden gem developed by Miyamoto that was locked in a vault. Although there aren't any "exclusive" Super Mario games