Forget about the wallets and coins.  Your thought experiment can be simplified.  Simply ask yourself:

"What is the most cost efficient and most effective way to uniquely identify a single person such that nobody else can pretend to be them and they can not pretend not to be themselves?"

Identification papers (passport, drivers license or national ID card, etc) can be copied, stolen, or faked.

The only answer I can come up with would be to require that the person be physically in your presence and that you find a way to reliably take a DNA sample from them.  Then fully sequence that DNA and store a hash of the result in a database.  Compare the results of each new person against all the results in your database.  Note that you may run into a problem with identical twins (triplets, quadruplets, etc).  In that case, you might require that they all be present together at the time of DNA extraction, and that if any are missing they will be excluded from the offer.

Obviously this would be a difficult and expensive thing to accomplish, but as a thought experiment it's the most reliable method I can think of.  You could reduce the costs if you were willing to accept that some people will be able to take advantage of the system.  How much effort and money you could save would depend on how much risk you were willing to accept of people gaining access to multiple wallets.