A Little History
You ask a very good and relevant question. In fact, back in 1958 H.E. Petschek wrote an interesting paper on "Aerodynamic Dissipation". In that paper, he hypothesized that one could, in theory, produce a shock wave in a collisionless medium (like most plasmas in space). This was highly controversial, since the very concept of a shock wave requires some form of irreversible energy dissipation in order for the structure to form. I wrote a few more details here and here.
Shortly after, Paul J. Kellogg predicted the existence of a bow shock around the Earth's magnetosphere. This was later confirmed by some of the first spacecraft observations. The controversy I mentioned earlier arose because a truly collisionless plasma is governed by the Vlasov equation (i.e., just the Boltzmann equation without the collision operator), which is a time-reversible equation of motion in kinetic theory. This is a problem because the formation of a shock wave requires energy dissipation (i.e., entropy generation and/or time-irreversibility).
As I discussed here, some form of energy dissipation is required to halt nonlinear wave steepening. Originally, there was debate as to how the Vlasov equation could supply any form of irreversibility. I wrote a response here that discusses irreversibility in more detail. None-the-less, shock waves can form in plasmas and are governed by the Rankine-Hugoniot relations.
Generally, for a shock wave to form one needs a piston moving through a medium faster than the relevant speed of communication. As I illustrated in this answer, there are multiple relevant speeds in space plasmas. Theory suggests one can get a slow mode shock wave to form in plasmas, but there is little evidence of those (at least of which I am aware). The more relevant speed is the fast mode or magnetosonic mode (Note: there are much better descriptions elsewhere on the web than the Wikipedia link, but that one was quick and easy). So the bow shock upstream of the Earth is a fast mode shock (thus the Mach number is defined with the fast mode phase speed), as are most astrophysical shocks to which you may be referred.
I am going to guess here, as I would need more information to answer properly, but I think your professor might have been talking about an asymptotic limit of the Rankine-Hugoniot relations for high Mach numbers. In the limit as the Mach number goes to some large number, the density compression ratio across a hydrodynamic shock goes to 4 (assuming a polytrope index of 5/3). One can make a similar argument for magnetohydrodynamic shock waves.