To use ' the ' in front of ' currency ' means that there is more than 1 currency in the US. Logically speaking, it is not right since this is not the case and only 1 single currency applies to 1 country only
Using ' the ' to the dollar to single out this particular unit of currency from among currencies used around the world. In fact, without ' the ' is still acceptable, though because there is no such strong necessity to do so.
eg. Dogs are a faithful animal.
eg. The dogs are a faith animal.
Both are acceptable, the only difference is that the second sentence used to
You can't omit the first "the" (or "a") in front of "unit" since "unit" is a countable noun in singular form. The same goes for "dollar".
There is no reason (at least none that I can discover) for or against a "the" before "currency". However, the common expression is "the unit of currency".
That said, the words "unit of" are expendable and may even be misleading.