I have seen two version of this;
The first version had a 3.5mm TRS plug extending from the base of each piece which plugged into a corresponding socket on each square. Each piece had two different resistors inside, and the CPU scanned these resistor values (as part of a simple voltage divider), determined the two resistor values and from that looked up the piece ID.
The second had a small battery, IR led and shift register in each piece that flashed out a serial byte encoded with a binary ID of the piece. Under each square was its own IR photodiode, detecting that stream of data for each.
By far the most common approach these days is machine vision. Painting the pieces in bold stripes and dots with IR reflective (clear) paint allowed a IR camera to easily see the various pieces. Most of these systems seem to have 2 cameras to compensate for blocked sight paths.
I wouldn't give up on NFR, but it will be challenging to get a cheap enough reader into each square, plus ensuring that they were so activated that they only read that square, and not the adjacent. Radio Waves are hard to confine
NFR/RFID is an option but not for very close range, and the readers are generally too bulky for you application