> Ideally currencies aren't supposed to have value per se,
According to whose ideals? Currency is a thing used as a medium of exchange. Like any thing used in trade, it will have value determined by supply and demand, both to individuals and in broad markets, and like all things that value will vary individual in ways that are subjective but in some broad patterns predictable (e.g., declining marginal utility is true of money as it is of anything else, though the fungibility of money suggests that the decline should be slower in terms of how much additional utility you can derive from money, compared to other things, before losing any given proportion of starting marginal utility.)
> You and I might disagree about the relative value of a product, but we express our own disagreement in terms of dollars.
That's not because we agree on the value of dollars ($/util) but because we agree to use dollars as a currency of account and/or medium of exchange.
> We don't disagree about the relative value of the dollar itself.
Relative to... what? I mean, the entire idea of mutually beneficial trade requires disagreement on relative value of the goods exchanged, and that doesn't change when one side of the exchange is dollars. So, yes, we probably do disagree on the most relevant relative value of dollars to the exchange, otherwise the exchange wouldn't happen.
I don't care how you value something (in terms of how much utility it gives you), I care how much utility it will cost me to get it from you. I don't need to concur with you on the value of the dollar to exchange dollars with you for goods.
> This is why inflation is usually measured in the shift of common goods and services in terms of dollars, since measuring the value of the dollar itself is a fool's errand.
Measuring what you can exchange something for is measuring it's (market) value, which is the only measurable (on a ratio level) kind of value. Subjective utility can be measured (with difficulty), but only to an ordinal level (x > y), and only for an individual.