In the textbook derivations of electron and ion plasma waves, the models of isothermal and adiabatic processes come from the assumptions in the flow of energy in the respective waves.

For planar, electron plasma waves, the choice of gamma=3 does not have much to do with "adiabatic compression". The planar nature of the wave gives N=1 to the dimensionality of the energy perturbation. Specifically, fluctuations in the electron temperature are only propagated in one direction, therefore N=1. It's often called adiabatic compression because of the assumptions necessary to eliminate the temperature fluctuations using the 2nd scalar moment of the Vlasov equation, or energy equation if you will.

In deriving the fluid dispersion relation for electron plasma waves, the linearization of the momentum equation yields an incomplete set of equations due to a term involving fluctuating electron temperatures. The temperature fluctuations are eliminated by considering the energy equation. One can only establish a relationship between the temperature and density and thus eliminate the temperature fluctations if the diverging heat flux, viscosity, collisionality, and convection are ignored. One then gets a relationship of the form dT=A(n-1)dN+B where A and B (usually 0) are time-independent and set by initial conditions, dT=electron temperature fluctuation, dN=electron density fluctuation, and n=(2+N)/N is the polytropic index corresponding to a polytropic process. This relationship completes the set of equations regardless of the actual value of n. In the above analysis, I've neglected essentially all forms of local heat transport and entropy generation, which is consistent with an adiabatic process and why it is said to be adiabatic compression. I am distinguishing n from gamma here because saying n=gamma is usually meant or reserved to indicate the assertion or determination that the process is adiabatic. For the ion plasma waves, however, the polytropic index is important, but the wave is not adiabatic.

For planar, ion plasma waves, the arguments are perhaps even less mathematically satisfying and rely on physical intuition because the interplay of electrons and ions is much more complicated and cannot be completely de-correlated as in the electron plasma wave. If the wave is now assumed to be propagated by the ions at frequencies comparable to their response time (inverse ion plasma frequency), we can expect the perturbations in the electron fluid to be somewhat blurred or smoothed out compared to the corresponding perturbations. Gauss's Law and the assumption of quasi-neutrality prevent us from de-correlating the electron and ion densities, but the fluid temperatures are correlated through secondary processes and can be de-correlated if we first make the same assumptions in the energy equation as we did for the electron plasma wave. Here, we get the same dT=A(n-1)dN+B, but independently for the ions and electrons as we cannot completely neglect either one. If the electrons respond so quickly that the temperature fluctuations are completely smoothed out, it takes on a value close to the average perturbation value. For an initially uniform plasma, that is dT=0. This is accomplished by setting n=1 and thus making the process in the electron fluid isothermal and de-correlated from the ion temperature. (otherwise, quasi-neutrality would connect the two temperatures) For the ions, they play the role corresponding to the electrons in the planar, electron plasma waves, i.e. n=3 for the ions in ion plasma waves. Usually, the ions are taken as much cooler than the electrons, so this choice is of little consequence.

That all being said, the above explanation is based on textbook derivations (my preference is Nicholson Introduction to Plasma Theory but Krall and Trivelpiece Principles of Plasma Physics also briefly discusses the choice of polytropic index). The value of n (or gamma) really depends on the validity of all assumptions mentioned above. Note that if the assumptions related to zero heat exchange are broken, the process is no longer adiabatic but may still be polytropic. In practice, the value of n (or gamma) cannot be set but rather measured or reasonably justified based on experiment.

In addition to the very completed and interesting previous answer, Dr. Purvi Kikani I would like to add that in MHD the process to take small perturbations is considered isentropic flow meaning that the perturbation in pressure field, delta p = cs2 delta rho with cs the sound speed. That is an adiabatic reversible process to describe magnetoacoustic waves, in a homogenous plasma.

See, for instance, Landau and Lifshitz, electrodynamics of continuous media, 1984, pp. 235, Pergamon.