Because thats how the buttons are mapped on the PS2 version of the Saturn Controller
L&R is old school mapping (mostly SNES and PS1 fighting games), R1 and R2 is new school mapping
For the Dreamcast you really want is a Saturn to Dreamcast converter with a actual Sega Saturn pad.
Most PS Fighters with L1 and R1 you can Map Moves to the Button
On this Adapter using the PS Controller with DC Fighters like SSF2X the L&R Triggers are Mapped to L1&L2 or R1&R2 on Joystick using the Sega Layout useless
Now the Adapter I linked is for SNES to PS Adapter…Hooking it up to the Innovation Adapter going to have the L&R Shoulder buttons on the SNES as the R Trigger on the DC Controller
It isn’t the adapter, its the controller that us map that way
For me to use it with Street Fighter II, III, IV and V, Dark Stalkers collection, Marvel and any other fighting game I had to change the input to put heavy Punch on L1 and Heavy kick on R1.