..surely it's only worth what it's worth because people say it is.
Yep. That's why anything in the world has worth. Or another word for it; value.
Gold. Art. Shovels. Chickens. They all have worth or value because we say they do. The amount of value something has relative to other somethings is based on only 2 things: how much of it that people want to own and how many are available to acquire.
But we can't confuse the words 'value' with 'usefulness'. We might say, "Fire has value because it keeps us warm and cooks our food." But fire doesn't really have much value at all. Yes, many people want to acquire fires, but the amount of fires available to acquire are practically infinite. While fire is 'useful' it isn't particularly 'valuable'.
On the other hand, art we might say, doesn't have much usefulness at all other than decorating an empty room or making overeducated intellectuals think deep thoughts where none are warranted. However I wouldn't go wiping my ass with The Monalisa if I could help it. Because it has value because people say it does. And there are only one of them.
So we can reason that value and usefulness are not connected. For example, if Leonardo had painted 21,000,000 copies of the Monalisa, it's value would decrease but it's usefulness would remain constant. Likewise, if there was only enough wood and flint in the universe to build one fire, it's value would increase, but it's usefulness would remain constant.
So yes fiat and Bitcoin are both valuable because people say they are.
Fiat money has USEFULNESS because the mafia.. *cough* I mean the government requires that you pay them off with it. For example, if I held a gun to your head and demanded your wallet, that wallet all the sudden has a huge increase in usefulness for you. Your life!
But does fiat money have value? Well yes. Because people want it and there is a finite supply (at any given moment).
Having said all that, I'm still trying to figure out why Bitcoin is useful and valuable. It breaks the rules. It seems to be a kind of paradox where it is only useful because it has value and it is only valuable because it has use. Basically, it's usefulness and value increase or decrease at the same time at a 1:1 ratio.
I guess it is useful because it can be used as a medium of exchange more effectively than fiat. Cheaper to transfer, easier to transport, arguably safer to store, predictable quantity, no middleman, anonymity, impossible to counterfeit, impossible to create more than 21,000,000 units.