That's fascinating! What a funny situation.
Is there actually any way to detect a counterfeit coin that is made of the right materials? It seems like all you've got to go on is the physical shape and relief. But that seems far easier to match perfectly than paper bills.
Is there at least any way to tell "old" counterfeits from modern ones?
Coin nerds study the real ones in excruciating detail - to the point that people catalog not only the various dies that were used to strike various coins but the states of those dies (i.e. Early die state == pristine coins. Middle die state == a crack or chip might have started to form which presents as a raised line or spot on the surface of struck coins. Late die state == chunks of the die might be missing.). This is often done through reverse engineering -- examining enough examples of struck coins to identify differences in the the dies used. The dies themselves can be identified through unique markers -- a polish line here, the exact positioning of a letter, a microscopic defect, etc.
So when you are able to identify the known dies, a counterfeit will stick out like a sore thumb. And if it was made from a cast copy of a known-good coin, then those will have their own telltale giveaways.
In short: The counterfeits will always have subtle differences.