Ah, there is additional hardware.
I'm not a hardware guy at all, so this might be wrong. But I *think* it does this:
The BIOS sets the output wire either high or low depending on which set of players it wants the inputs from. The board reads those controllers and returns the corresponding inputs. Those inputs come into the Neo via the same REG_P1CNT/REG_P2CNT as any other game, but the BIOS knows which player they are for as it set the output line, so it copies them off into memory, where then Kizuna Encounter can read it.
If that is correct, it would only read inputs at 30fps, which is not great for a fighting game, but probably still good enough as this is really just a novelty.