Since the speed of sound depends on the air, which changes at different altitudes, are “Mach” speeds dependent on the altitude? Or is it basically just like 100 °C is the same even at altitudes where water boils at a lower temperature (in that it is a fixed speed based on the speed of sound at sea level)?
Yes, because its calculation relies on the local speed of sound, which is varies based on several factors, including altitude.
https://en.wikipedia.org/wiki/Mach_number