and each dollar has the same value

The intrinsic value of fiat currency is 0. Double, halve, quadruple 0 all you want makes no difference. It's function and value is as a medium of exchange.

Imagine a copper based currency. If supplies of copper increase, the intrinsic value of copper falls, so the total value of the currency falls. The extrinsic value is not affected.

If I buy a widget for $1 and my labor is $2, I can be paid in 2 widgets. The money supply doesn't change that my labor is 2 widgets. If prices are increased on widgets by a capitalist, then I would expect an increase in my labor price (in dollars), regardless of the money supply, because money has no intrinsic value.

I'll state again that this difference (capitalists choosing to raise prices vs blaming external factors like "money supply") is not just pedantic. Capital mentions it few times, the fetishization of money and capital accumulation/hoarding cause this belief that money has a function outside of exchange.

To put it mathematically: the rate of accumulation is the independent, not the dependent variable; the rate of wages is the dependent, not the independent variable. Thus, when the industrial cycle is in its phase of crisis, a general fall in the price of commodities is expressed as a rise in the relative value of money, and, in the phase of prosperity, a general rise in the price of commodities is expressed as a fall in the relative value of money. The so-called Currency School* conclude from this that with high prices too much money is in circulation, with low prices too little. Their ignorance and complete misunderstanding of the facts are worthily paralleled by the economists, who interpret the above phenomena of accumulation by saying that in one case there are too few, and in the other, too many wage-labourers in existence.