We'll see what others come up with, but it seems to me like the best existing word available might be "unit," as used in the question.
OED definition 3.a. for dollar describes it as a "unit." Essentially it is a "unit of currency."
3.a. The standard unit of the gold and silver coinage of the United States of America, containing 100 cents.
But for some time, the word dollar functioned in English as a generic term for currency, to some extent. First named after a foreign currency, the German thaler, "dollar" would be applied to various newly encountered currencies, first with the Spanish peso, and later with others, as outlined in OED definition 4.a.:
4.a. Also used as a name for various foreign coins of a value more or less approaching that of the Spanish or American dollar; as the peso of Mexico, and of the republics of Central and South America, the piastre of Arabia, the yen of Japan, etc.
Perhaps that's one reason why we don't have another word in English for a unit of currency regardless of type. But it also makes sense, because most of the things we need to say about money can be said by discussing "currency" and the names of currencies. When we need to be specific about the physical representations of the currency, we can use "coin," "bill," or "note." When someone in the U.S. says "hand me a dollar," they could also say "hand me a dollar bill." In other countries they might say "a dollar note.". In "hand me a dollar bill," "bill" is a generic term signifying one piece of the currency, and "dollar" is the currency itself used as a modifying adjective.
For most of the history of the English language, the word "coin" would have been close to what you're looking for. A coin was one "piece of currency" regardless of what currency it was in. Hence, in the definition above, OED refers to a dollar as "a name for various foreign coins." So, prior to the adoption of paper money in the 17th century, it would have been the ultimate hypernym for one piece of any currency.
