Prices are set by supply and demand, which you may choose to label as "power" as a matter of a priori definition, and to some extent that will be a suitable label, but you would be throwing a way a whole deal of explanatory power by insisting on such a rhetorically loaded framing.

> How often is "You should pay me much more, it will make both of us wealthier" a winning argument when asking for a raise?

A great deal. If you don't pay me, I will quit, and you will be worse off, so it's a win-win proposition for you to give me a raise. That's what labor market competition is about.

> They end up with a price because they're mediated in currency, but their differences are far more interesting and economically revealing than their very superficial similarities.

Hence microeconomics, labor economics, financial economics, and the various other mainstream economics disciplines that do try to split those hairs.