I have an implement for doing exactly that on my desk - a cup. The people who made the cup don't know what I'm going to be filling it with but they have a very good idea of what volume it'll take up. I could go to the pub with friends and ask for a glass of something but I don't mind what. The point of a unit of measurement is it enables abstract handling of quantities. Otherwise we may as well have a unique system of measurement per thing.

And if you want a monetary example, there is barter. I can exchange $50 of work directly for $50 of food, abstracting out the money. That wouldn't be possible if the token itself were the important thing, because it isn't present anywhere in the example.

> Fiat money is a token of exchange backed up by the need for everyone in an economy to pay taxes using those tokens.

Obviously there is more than one type of money if you feel a need to add a prefix to explain what type of money you are discussing. The other types aren't backed up by a need for everyone to pay taxes, money can exist independently of a taxation system. Then it is called non-fiat money. You're focusing on the non-monetary aspects of the system, which is cool and all but missing the forest for the trees.