the Austrian School generally treats money as a commodity, it's price measured by its purchasing power.

if people have higher demand for money, it would naturally be more expensive as measured by other goods.

No, demand is finite in price theory. It has dimensions of quantity, price, and a curve moving along different quantities & prices. It also has many inputs that can be quantified by elasticities.

It is the individual's desire that may be considered infinite, but even them we have ordinal numbers to determine what desires are stronger.