In both games, getting hit in Story mode will give the player character battle damage in future scenes. These are the four limbs and either the head (BB1) or torso (BB2).
Here's those Mikado scenes from the first game, but this time both Mikado and the other playable characters are fully damage. I only recently discovered you could even apply the damage effects to NPCs. It's an exploit anyone can do: using the secret P2 option in Story mode (R1+R2+L1+L2+Start+Select almost anytime on P2's controller while in Story mode), any NPC in P1's story that is defeated while controlled by P2 will be fully damaged when they reappear. No gameshark codes or other data manipulation required at all.