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Mach number is a ratio, not a fixed speed: it is the true airspeed divided by the local speed of sound. The speed of sound in dry air depends only on temperature, following a = √(γRT) with γ = 1.4 and R = 287.05 J/(kg·K). That is why Mach 1 is about 1,225 km/h at sea level but roughly 1,062 km/h in the cold air at cruising altitude — the same Mach number is a slower true airspeed when the air is colder.
Speed of sound
a = √(1.4 × 287.05 × T) where T is in kelvin
Mach number
M = true airspeed ÷ a
Because the speed of sound falls as air gets colder. Mach 1 is 340.3 m/s at 15 °C but only about 295 m/s at −50 °C, so the same Mach number means a lower true airspeed high up.
Not on its own. For an ideal gas the speed of sound depends on temperature alone — pressure and density changes cancel out — which is why only temperature is needed here.