What does Mach speed mean for describing aircraft speed if the speed of sound changes with temperature?
The aerodynamics change with Mach number.
At low Mach numbers (< ~0.3) , air is considered incompressible; it has constant density along the stream line.
At higher Mach numbers, air is no longer incompressible; the density changes as the air passes over the aircraft. The compressing and expanding of air changes the way the air flows, and consequently it changes the performance of the aircraft.
At Mach numbers approaching 1, parts of the flow around the aircraft may breach the sound barrier. This leads to trans-sonic effects such a shock waves which affects the performance of the aircraft even more.
For subsonic aircraft the maximum speed is typically expressed by two numbers: the maximum calibrated airspeed and the maximum Mach number. The first is related to the dynamic pressure the flow of air creates and is typically the limiting factor at lower altitudes. The Mach number is related to trans-sonic effects and is typically the limiting factor at higher altitudes where are is colder and the speed of sound is lower.
For the pilot it thus important to know both the airspeed and the Mach number.