On analogies between gas and stellar systems Analogies between (typically) ideal gas and stellar systems are not only intuitively valid to some extent, but have been established and used in the studies of stellar clusters and galactic systems, most often as a simplification to collisionless Boltzmann equations.

The idea behind the analogy is that if a stellar system can be represented as a set of point masses, and if the number of point masses is large, then we can consider them from the point of view of kinetic theory of gases. One thing to remember here, though, is that stellar gas system is neither relaxed, nor can be relaxed.

I am curious here: How far can the described analogy be pushed?

For example, there is a range of gas-specific phenomena (or we could be talking about plasma, if you may prefer), which would be fascinating to imagine for stellar systems, such as shocks, turbulence or viscosity. Can such, or some other, characteristic phenomena exist in stellar systems and are there actual systems exhibiting such a behavior? (of the named ones, viscosity analogue exists and is rather common)
The analogy is rather weak and not really useful.

So-called collisionless stellar systems (those for which relaxation by stellar encounters has no appreciable effect over their lifetime), such as galaxies, can be described by the collisionless Boltzman equation, but never settle into thermodynamic equilibrium (only into some dynamical or virial equilibrium). Thus, the only other systems with somewhat similar behaviour are collisionless plasmas.

Sound, turbulence, viscosity etc are all effected by close-range collisions (not mere encounters) between the molecules. These also maintain thermodynamic equilibrium and a Maxwell-Boltzmann velocity distribution. Stellar systems have none of these processes and their velocities are in general anisotropically distributed and don't follow a Maxwell distribution.

Gases are in some sense simpler to understand, because their dynamics is driven by local processes and because statistical methods are very useful. Stellar systems are driven by gravity, i.e. long-range non-local processes, and intuition from the physics of gases is often very misleading (for example, a self-gravitating system has negative heat capacity -- this also applies to gas spheres, such as stars).

Note also that the number of particles in a gas is much much larger ($\sim10^{26}$) than the number of stars in a galaxy ($\sim10^{11}$), though the number of dark-matter particles may be much higher.