How to choose the correct inductor for my application
It seems you don't want inductors at all.  You apparently want to transfer power over a gap using magnetic induction.  That's what a transformer does.  The game board will have a bunch of transformer primaries, and each game piece will have its own transformer secondary.

You're not going to find parts for that off the shelf.  You'll have to create your own windings, possibly with ferrite cores of some sort to better concentrate the magnetic field so that most of the flux goes thru the loops of the coils.

This is nowhere near as simple as deciding what inductor to use.