Do monetary aggregates measure money demand?

I don't know. How's that for a beginning? If we take the currency in circulation (MBCURRCIR at FRED aka 'M0') to be the supply and either M = M2 or MZM to be the demand, we have the solution to the differential equation M = a M0^(1/κ) (first plot, money demand in terms of supply) or it's inverse M0 = b M^κ (second plot, money supply in terms of demand) ... these result in decent, but not amazing fits:

One pedagogical point: in this case the units on each side are the same information-wise (dollars), hence κ doesn't vary when demand is dollars like it does when demand is "dollars of goods and services" (NGDP). Also note that κ ~ 1 in both cases implying that the "price" of money P ~ M0^(1/κ-1) is roughly constant. Another way, the "exchange rate" between demanded money and supplied money is constant.