Money is an abstract idea, so how could some government/king come up with it?
Well, let's say you owe someone 3 chicken. So what? It's irrelevant as long as there's no law enforcement behind it, which can only come from authority. /cc people piraped at 40.
Ancient communities, just like American natives, didn't think in terms of barter, tit-for-tat. They mostly shared among themselves what they hunted and gathered. This is also called primitive communism.
What you could argue is that this model didn't stand the test of time. The evolution of mankind favored leaders, conquest, authoritarianism. And I guess Graeber wouldn't deny that a monetary system indeed may allow more efficient allocation of resources than a free sharing model. But he insists that money was an authoritarian invention, to collect taxes and support soldiers.