I think you are misreading the partial schematic.
Each common anode layer needs 1 wire.
The common cathodes need 64 wires, 1 per column, usually soldered to the PCB that makes the base of the cube.
72 wires total.
Vs 8 x 16 wires = 128, and connectivity not built into the array either. See the video that was posted in Reply #21.
Wish I had way more hours in a day to build one of these up.
I'd go for 9x9, have read having the center LED to pivot things around makes it look neater.
My 12-shift register board could drive it, be a bit of a software PITA spreading 81 bits + 9 anode control bits across 12 8-bit registers (96 bits).