I know that's a rhetorical question, but still...
Essentially, "backing" in the context of currency is a promise by the issuer of the currency that it is at any time redeemable for something that its users consider valuable, and that the exchange rate stays constant.
As with all promises, this promise can be broken - intentionally or unintentionally. For example, if I issue a currency that is "backed" by huge warehouses of food, its users might consider it valuable (and use it for their businesses) because they know that at the end of the day they will get enough to eat for their money. However, if the food gets spoiled or stolen (not intended by me), the money will become worthless. Another possibility is that I print more money (because I want to buy services from the folks using it) without adding corresponding amounts of food to my warehouses - then we have inflation, and the money becomes worth less over time.
Back in the time when money was "backed with gold" the common understanding was that a certain amount of money could be exchanged for a certain amount of gold. Since gold is considered valuable by most people (which is entirely subjective itself, but that's a digression) that worked pretty well most of the time.
However, nowadays the currency used in an economy is typically not backed by some static resource such as gold, but by the value of the economy itself. Governments promise that you will be able to get goods and services within that economy, and they (partially) try to keep their promise using their monetary policies. However, money stability isn't their only goal, there are other sometimes conflicting economic goals (such as employment rate and growth) which they try to achieve as well. As a result, inflation is often controlled to be high enough that investments are attractive, but not too high so the citizens won't riot :-)
Of course, this leaves the impression that "your money" constantly loses value, and that the government does a bad job of keeping its value constant. This is a misconception, it's not your money, it's their money, and it's working as designed. Note that most of the wealth of the truly wealthy is not kept in the form of cash or bank accounts, but in the form of assets which gain value (or pay dividends) over time.
I think that's the main problem with fiat, and not the question whether it is backed by anything. That question is a red herring, because it only leads to a recursion.
Instead, I agree with the OP that value is entirely subjective (although there are objective needs such as food and shelter, they can have wildly varying costs) and that the concept of "backing" is not very useful.
Onkel Paul