This classification of money arose in the attempts to explain the value of money. The value of a consumer good comes from the aid it gives in directly satisfying a person’s end. Money does not directly satisfy a person’s end. The value of money is in facilitating exchange. The value of a producer good comes from the aid it gives in producing consumer goods that directly satisfy a person’s end. Producer goods are used up in production. Steel becomes a fender for a car, labor services are used up in assembling the car and so on. Money is not used up when rendering its services. It trades from one person to another to another with its physical integrity intact.
The other issue you raise is the item used as money. Let’s start with fiat paper money. In that case, it should be clear that its value cannot come from it being either a consumer or a producer good.
In the case of silver or some market commodity, people will arbitrage the stock of silver across all the different uses that have sufficient value so that the value of a unit of silver is the same in each use. Even so, the value of a silver coin is determined by the value people place on it as a medium of exchange. Just as the value of silver spoons is determined by the value people place on silver spoons as a consumer good.
This does not create an overlap of categories of goods, but a diversity of classification of a single object (silver) according to its uses (consumer good, producer good, medium of exchange).