If the item is shipped by a third party directly, tracing the counterfeit origin is easy.
It's the comingling at the Amazon fulfillment centers that muddies the water. However, why not just tag each individual item with the supplying vendor as it arrives? You must of course credit them with stock, so just assign a code for that unit or box or pallet connected to that seller. They can still cross-fulfill (though I don't think they should) and record the origin code with each order as it's packed. Don't tell me this is too complicated for Amazon. It's an extra code like a second SKU per item, not conceptually challenging. Of course there are process issues to solve, but this should be a huge internal priority for Amazon. I see it as a weak spot that leaves room for a competitor to do better.
Of course identifying fakes is still a challenge, manufacturers have long had systems for verifying authentic units. If a $20 webcam has a unique serial number I don't see why we couldn't have one for an $80 board game.