Money is a socially acceptable and countable medium which is used to quantify the exchange of goods, services social credits and obligations. A monetary system is a group commitment to the value and rules of usage of an ephemeral convenience which is used to facilitate the distribution of resources and the correction of imbalances.
There is a group of essential properties which money needs to possess in order that it may function satisfactorily within a monetary system. It must be socially trusted and accepted...individuals and groups must have confidence that the value promulgated or claimed can be relied upon. It must be precise and unambiguous in its countability...different sized goats or unequal bags of grain will eventually be very unpopular. It must be durable, stable, resistant to rotting, decay and inflation...trust is seriously undermined if banknotes disintegrate or the digital memory locations of a bank balance are erased by a software glitch. It is very desirable that the money be portable and compact and of high "density"...tourism would be impossible if the monetary means of exchange was camels. It must somehow be extremly resistant to counterfeiting...the cloning of the medium and its injection into the market by self-interested individuals. Such action devalues the medium and undermines the essential element of public confidence.
In order to fulfill a wide variety of economic functions money must be obtainable and of a suitable form . As a minimum requirement, it must be a satisfactory medium to transact all the various forms of payment, lending, compensation, gambling and gifting that have evolved as societies have complexified . Money is an essential mechanism to partitioning cooperating and evaluating complex human social endeavours. The construction of any extensive engineering undertaking (such as buildings, aircraft or nuclear power stations) or the organization of public institutions (such as education and health), require that the various and complex elements be allocated proportional monetary values in order that the projects become generally acceptable, and the diverse particpants are satisfied with their level of compensation.
The social credits of "money" are only potential wealth, and it seems to be distorting and manipulating ideas to promote the suggestion that money, as potential wealth, is also actually wealth. What is potential is not actual and never can be. The "money" of one society is often devalued or even not recognized by another society, so the potentially wealthy in the first is no longer so in the second. Money acquires "value" by being exchanged for something. Wealth is measured by the quantity and quality of resources actually being used. Burning money is not a measure of wealth but a measure of wasted resources.
The ownership of money must be claimed by a central administering authority and distinguished from "possession". There would normally be some sort of central agency that acts as the monetary resource for the system. A king, for example, or a governmental agency of some sort needs to assert and police an ownership of the money. The agency would in effect create a pool of credit for itself, and then use that credit to advantage itself or to advantage any others that it chooses. Thus to introduce a new monetary system, a king could acquire food and resources "free" by paying for them with money he had created, and thus facilitate the circulation of the money into the society at large. He could also lend his money out to those who had none, so that they may use it as they wish, but with the expectation that it would be returned with interest after a period of time. Elected governments could behave collectively in a similar manner, where some of the purchases..like schools or hospitals...would be used by the society which elected them. In the present era however, new governments normally inherit an existing system, and so are usually constrained to raising funds by taxes and manipulating the rules of participation in the system. The central agency could be thought of as "owning" the money in the sense that they must claim the right to create limited amounts of it, and exercise their power to legislate and enforce exactly how it may be used. It must sanction its own set of rights over the money, and authorize itself to sell or exchange it in the same manner in which ownerships are determined in general.
Media which have been used by societies as money, have evolved from utility barter items such as food, manufactured artifacts, and valued environmental finds, via specifically created durable objects such as metal coinage and fragile printed paper assertions of value, thru to the present digital configuration of electronic components. The capacity for fraud has increased by several orders of magnitude thruout this evolutionary process simply because the capacity to generate what counts as money has become easier and easier to fabricate. A medieval king had to have actual coins minted to make more money, but modern digital, accounting and legal manipulations can manufacture numbers as money where none existed before. More and more individuals are now able to contrive numerical quantities into the records of their monetary credits, without having contributed anything whatsoever to the wealth of the society.