Ah, that is different.
Well I have designed a board with 18 telecom type relays,
you can see a picture at the link in my signature
that could easily be expanded to 56.
Your need is to have one (or more?) banks of relays - only 1 bank at at time would be selected to be connected to the cabinet.
Relays have a Common, Normally Open & Normally Closed contacts. The Common for each line would be connected all in parallel to the cabinet, with the individulal boards connecteed to the Normally Open. When a game is selected, the Normally Closed contacts would be connected.
That's certainly doable.
Not sure where the 4 player 8 button breakout comes into play.
Not sure where the Arduino comes into play. Hardly seems necessary to enable one bank at a time.