The same pertains to pure fiat (not backed by anything). Actually, that was one of the reasons it came into existence and got universal acceptance (exclusion of externalities, such as finding a gold asteroid, for example).
Fiat like USD doesn't need to be backed by anything. Its legal tender so by law you have to pay your taxes in fiat. Its the only acceptable form of payment. The amount of taxes are correlated to GDP so you are correct. The price (compared to other currencies) fluctuate on economics
Because its mandated that people have to pay taxes w fiat, the demand is always there. This guaranteed demand is what makes USD stable. As long as US economy is stable
In a broad sense any money is backed by the utility it provides, either as a payment of taxes or as a means to purchase goods and services (including a utility for speculation, of course).