An economy is understood as a system of exchange where values move in a forward and backwards fashion. For instance, it can move from buyer to seller and back. The current monetary system does not accomplish this goal.
What makes a monetary economy
Essentially a monetary system cannot function without a governing body which oversees the flow of money. An association that advocates for certain selfish values can be formed by a small group of people and if it gains enough viral traction from a larger set of members it can impose the values as rules upon the remaining minority. This is what happens with the monetary system. Those who were in possession of political power decided to disseminate the idea that they have the responsibility to provide a uniform representation of value (money) straight out of the blue. The responsible body is known as the Central bank or, in the U.S, the Federal Reserve.