I own this adapter. The saves go in a PS2 memory card folder just like for the PS1 saves. But if they don't work with PS2 Classics then it's not much use, then.Someone on reddit pointed out there's a USB memory card adapter that you should check your local game store for if you're super serious. Amazon prices apparently are ridiculous.
Beyond that, I have no idea where you'd put the save to make the game recognize it, soooooo.... :\