Well, that's correct, so far as it goes...but it wasn't quite what I was looking for.
You see, you can say (for example) that one degree Celsius is "a unit of temperature," but to be specific you say that one degree Celsius is "one one-hundredth of the difference in temperature between the freezing point of water and the boiling point of water at sea level." Likewise, one gram is the weight of a cubic centimeter of water, and one milliliter is the volume of that quantity of water. A liter of water is the volume of water that weighs one kilogram. The units are defined in terms of measurable quantities of real materials with fixed properties.
In other words, the units used to measure things are defined in terms that allow us to quickly understand their meaning. When we were on the gold standard, the dollar was also like this. Instead of saying that a dollar was "a unit of fiat currency," you could say that it was "a unit of currency equal in value to X weight of gold."
Since we're off the gold standard, that's no longer a valid definition...X isn't fixed any more. So what's a dollar? It used to be influenced, if not controlled, by the relative abundance or scarcity of the physical paper notes that represent dollars. Now, however, we don't even remotely have enough paper notes to represent all the dollar transactions that occur in a given day. A number in my bank's "ledger" goes down, a number in my utility company's "books" goes up, and no paper notes are involved.
So what's a dollar again? How is it defined? What determines its value?
Yes, I know your next question, the value of the dollar is an abstract. Technically, however, so is the value of Gold (or anything else for that matter.) in that the value is purely related to demand, demand comes from a perceived value.)
I will jump to the chase :) A dollar has no intrinsic value. Its value is determined by supply and demand. The supply is determined by the reserve rate and the demand is determined by the GDP and interest rate.
Your analogy to units of temperature fails because those are measurements. Your analogy would be more appropriate to the interest rate determining the value.