None of tsuwm's references actually explain why. They just give dates of first usages and examples of usages.

I've googled around a bit, and I can't find a historical perspective of how the word came to be used in terms of money. But I can probably guess.

If you think of money as the oil which makes the economy run smoothly by providing an efficient medium of exchange, it's probably easier to grasp the use of the word. "Currency" is probably a contraction of the term "units of current value". Money remains money, but the value of your money is changing all the time, even while it sits inert in your wallet. Money is worth what the market will pay for it at the current time, either in other currencies or in exchange for goods. Therefore the "current value" of your money is different from the "face value" of your money. A dollar is always a dollar, but the real value of a dollar is measured against the value of other currencies and, by extrapolation, against the goods you use your dollar to purchase.

The term "currency" only applies to negotiable instruments, i.e. monetary instruments such as notes, gold, silver, platinum, coins, cheques, drafts, etc., which can change their overall value. It doesn't apply other mediums of exchange such as barter or direct exchange.

Hope this is clear.