What are the aerodynamic implications of different conditions with the same Mach number?
The Mach number only indicates the behavior of compressibility effects, but does not provide any indication about other effects like inertia and viscosity (read turbulence) or heat transfer.

For high subsonic and supersonic aerodynamics one should typically compare both the Mach number (compressibility) and the Reynolds number (turbulence).

So at the same Mach number but different densities, the Reynolds number will be very different and therefore the turbulence effects will differ highly. Only the compressibility effects will be similar (shock waves).

Note that at low subsonic speeds (Mach&lt;0.3), the compressibility effects become negligible and one can get away by only comparing the Reynolds number.
And at hypersonic speeds (Mach~>5) more similarity parameters are needed.

More information about dimensionless numbers in fluid mechanics on Wikipedia.