Ultimately, the speed of sound in air depends on the air temperature only.
In fact, assuming an ideal gas, the speed of sound c depends on temperature only, not on the pressure. Air is almost an ideal gas.
Therefore, you can have the speed of sound be different for the same density and pressure, as long as the temperature is different. Density and pressure determine flight level and TAS-to-IAS, so you'd be at the same FL, the same IAS, the same TAS, but a different Mach number.
In a single volume of gas, of course, I can't change JUST T and not p and V. But we're talking about a case of moving from one section of the atmosphere to another, so the gas law relationship doesn't have to hold.
So the original text is correct - if you pass a front such that there is a marked temperature change, you may find the relationship between Mach and IAS changes, even at a fixed FL, due to temperature changes.