>How did the word.. come to be applied to money?

yes, this does seem to be the salient question..

3. Of money: The fact or quality of being current or passing from man to man as a
medium of exchange; circulation. Also fig.

1699 LOCKE 2nd Reply to Bp. of Worcester (R.), 'Tis the receiving of them by others, their very passing,
that gives them their authority and currency. 1722 Lond. Gaz. No. 6078/2 All such of the said Bills..lose their
Currency. 1729 POPE Dunc. I. 23 note, The papers of Drapier against the currency of Wood's copper coin in
Ireland. 1862 RUSKIN Munera P. (1880) 15 The laws of currency and exchange.


4. a. That which is current as a medium of exchange; the circulating medium (whether
coins or notes); the money of a country in actual use.

1729 FRANKLIN Ess. Wks. 1840 II. 270 Money..by being coined is made a currency.