As I understand what you really want to measure is not proximity but whether a spot is occupied or not, right?
Photodiodes are actually analog sensors, but can be read as digital depending on the way you wire them. Same for other light sensors such as LDRs. Alternatives may be magnetic sensors such as hall effect sensors or reed contacts, triggered by a small magnet in the bottom of the game pieces.
Shift registers are probably indeed the cheapest, simplest solution to read such a number of sensors.