I think it comes down to the ideal gas assumption. In an ideal gas gamma is constant and for a diatomic ideal gas (two atom molecules) the theoretical value is 1.4.
But of course an ideal gas is just a theoretical model. It may be a close enough description of real air for many purposes, but it is not exact. To the extent that real air is not a true ideal gas gamma may vary a litttle bit with temperature.
Also as you point out air is not entirely diatomic (N2, O2). It also contains H2O, CO2 and traces of monatomic gasses which will affect the gamma value.
Nevertheless flight instrument calibration equations are based on ISA which assumes inter alia that air is a ideal gas and that gamma = 1.4.