In the interstellar plasma, objects moving faster than the Alfvén velocity also create bow shocks.
No, the obstacle must move faster than the fast/magnetosonic speed, not the Alfven speed to generate a shock wave in a plasma.
By analogy, one might imagine that such objects would also experience wave drag. Is this a large effect, relative to drag from direct collisions?
As I point out in this answer and this answer, the Coulomb collision rate is absolutely tiny in the interstellar medium. The time necessary for a charged particle to cross a collisionless shock ramp is on the order of milliseconds whereas the mean time between Coulomb collisions can be days. So no, collisions do not cause drag in the sense I think you are asking on shock waves in most plasmas, thus why they are called collisionless.
However, collisionless shocks do dissipate energy through other means like dispersive radiation etc. (see this answer for more details).
How could one estimate the magnitude of the drag force?
You would need to measure all the energy inputs and outputs of the system to do this properly, which is precisely what many researchers attempt to do with in situ data [e.g., see Wilson et al., 2014b; Wilson et al., 2020; Madanian et al., 2021 and references therein]. The dissipative effects are precisely what will reduce the currents in the shock ramps, which will reduce the magnetic field gradients, which will eventually result in the collapse of the shock back into a normal fast/magnetosonic wave. This wave will continue to damp by interacting with the charged particles in the medium and eventually be reduced to a thermal normal mode of the system in the absence of a free energy source.
The currently accepted answer to Would a fast interstellar spaceship benefit from an aerodynamic shape? states that an interstellar spaceship designer would not have to worry about the sort of streamlining used in airplanes, because the length scale for turbulence cannot be shorter than the mean free path of the gas molecules. In this case, that's on the order of 100,000 km. However, I don't think this is a convincing argument against the existence of wave drag. Most shock waves in astrophysics are collisionless, meaning that they can exist on length scales much shorter than the mean free path. For example, the bow shock created by Earth's magnetosphere is only about 17 km thick.
Yes, shock waves exist on spatial scales much much smaller than the Coulomb collisional mean free path of particles. In the solar wind the anecdotal statement is that the collisional mean free path of a thermal proton is one astronomical unit while the spatial scale of a collisionless shock ramp is less than an ion thermal gyroradius (e.g., ~50-150 km near Earth), i.e., six orders of magnitude or more difference. However, as I stated above, collisionless shock waves, and any other plasma mode for that matter, do not dissipate through Coulomb collisions but through other electromagnetic processes. These are not relevant to to an interstellar spaceship (except maybe charging and resulting arcing but that's easily avoided by using highly conductive outer surfaces). That is, an interstellar spaceship will not generate a shock wave unless it is highly magnetized (i.e., it creates its own magnetic bubble, if you will) and moving faster than the local fast/magnetosonic speed.
The currently accepted answers to Do celestial objects experience drag from the near vacuum of space/does the near vacuum have a mean velocity? and Interstellar medium shock heating do not consider wave drag either.
That is because the type of drag you mention is not relevant to these scenarios, as I explained above.