It's exactly the opposite. With languages/standards, both parties has incentives to have mutual understanding so they don't need anyone else to enforce symbol equality. But with these weights, or money in general, there's huge incentive to deceive each other, so someone has to enforce the equality. That someone can be the parties themselves, but if one party lack the ability, it necessitated the creation of third-party enforcer, which can grow to be a state.
Even then, with languages, whenever there's incentive to deceive it also immediately unravels. See: exaggeration, and necessity to create whole new language of legalese for contracts.
Then there's this theory that legalese was created to enforce the need for legalese, to justify it and feed it forward.
You're talking about a time pre-capitalism. "Incentives" don't map to a stable cross-community market. We should talk about "reasons," not "incentives." A great reason not to try to deceive about weights is it would make nobody wants to trade with you when it gets out that you're dishonest. It may have also been considered deeply unethical in their society to do so, there doesn't need to be any enforcement mechanism beyond that for it to manifest across the entire society in an archaeological record.