Overview#Unit of measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity.
Any other quantity of that kind can be expressed as a multiple of the unit of measurement.Medium of exchange in use, e.g. coinage should be in denominations of that unit making accounting much easier to perform, it has often been the case that Medium of exchange have no natural relationship to that unit, and must be 'minted' or in some way marked as having that value. Also there may be variances in quality of the underlying good which may not have fully agreed commodity grading.
The difference between the two functions becomes obvious when one considers the fact that coins were very often 'shaved', precious metal removed from them, leaving them still useful as an identifiable coin in the marketplace, for a certain number of units in trade, but which no longer had the quantity of metal supplied by the coin's minter.