As an exercise, I worked through a rather similar example a while back, using performance charts from a C172 POH.
===
This C172S has an IO360 180bhp engine, and a fixed prop. Data is from the POH.
The 2,000 ft performance figures for 2380 RPM are interpolated from the 2300 and 2400 data. The point was to match KCAS at different heights.
One would expect wing, propeller and engine performance (specifically nautical miles per US Gallon of fuel) to vary in different ways with the lower air pressure and lower air density at altitude. This analysis chooses 2380 RPM at 2000 ft so that the CAS (Calibrated Air Speed) matches the CAS at 2700 RPM and 10,000ft. In this way the Lift Coefficient and so the wing performance should be the same in both cases, in particular the Lift/Drag ratio. As Drag times Distance gives energy expended, the mpg figures should be the same in each case, if the wing performance is all that matters.
However, there is no obvious reason why energy losses from propeller and engine inefficiencies at altitude should depend on the Lift Coefficient or CAS, and these should generate extra Drag, and make the mpg figures differ.
The table shows that the mpg figures do match! Also, as if by magic, the key propeller coefficient (the Advance Ratio), and one key engine statistic, the BMEP (Brake Mean Effective Pressure) are the same as well. Practically all the other measures vary by the ratio of the air densities.
The conclusion is that range performance depends only on CAS. Even the propeller and engine efficiencies depend only on CAS (or at least their combined efficiencies do). It is not clear to me whether this is some grand principle at work, or "just" clever design by Cessna.
===
It pretty much backs up what Big Pistons and mm_flynn have said already.
One point not made so far: for a given CAS, the TAS is higher at altitude, so for a fixed-pitch prop to have the same angle of attack, the RPM has to be higher. RPM and blade length are not affected by air density
.
The table shows this as the "Prop J" aka the "Advance Ratio".