The relation between Mach number and TAS is a function of temperature.
The relation between Mach number and total pressure / static pressure is NOT a function of temperature. Since a Mach meter works from inputs of total pressure and static pressure it is not theoretically subject to errors due to deviation of SAT from ISA.
This leaves open the question of why early Mach meters suffered from temperature errors. Maybe temperature errors were caused by expansion / contraction of the mechanical linkages. This would seem to be more related to cockpit temperature rather than ambient ISA deviation as the original quote implies. Also it does not explain why a modern air data computer would need to correct for temperature errors, assuming the pressure transducers (ADMs) dont have significant mechanical linkages.