Hmmm...my Logitech Cordless Action wireless controller works fine on my fat PS1, I use it all the time.
PS2 controllers function identically to PS1 dual shocks when used on a PS1, so games that don't support analog will not enable analog functionality on the controller and should still work fine.
Also to the original OP, technically I don't believe there's emulation involved in the PS2's ability to play PS1 games, as I believe that the major components of the PS1 are in the PS2 in hardware form.