What's the relation between this and MAME? I don't have the Bomb Jack arcade PCB (I've got others though) but I do have a Pi2JAMMA and, well, Bomb Jack using MAME.

Is this project something that could be replacing (partial?) hardware on a real Bomb Jack PCB (not unlike what some are doing with C64 chips, where new replacement can be dropped in place of old broken chips)?

It started off as a direct Bombjack arcade hardware replacement, it then grew way beyond what the arcade was capable of. MAME is software, this is hardware, no relation. :) Although interestingly the first version of the hardware did allow me to find a bug in the MAME implementation.