Since I am sure you're not denying the authenticity of this video, let us turn your question another way: why is the flag behaving this way ?
The flag is roughly subjected to 3 forces :
- Its weight: 1/6 of the very same force it would have been subjected to on Earth
- Tension from its attached part on the rod
- Internal material tension due to the elasticity of the fabric
Pushing the flag into the ground, Armstrong gave it an initial momentum, and the flag was then only subjected to those three forces in the nearly vacuum of space. Please note it is a very classical physical situation: what we have here is a pendulum with a mass (the fabric) a tension (the fabric is attached to the rod), a fixed part (the rod), and an initial velocity. What many have wrongly interpreted as some wind proving the scene would have been recorded on Earth is just the oscillation of this pendulum. Such a scene is impossible on Earth since with such a low mass as a pendulum, friction forces from the atmosphere added to the other three force would impede any oscillation movement.
Now about Mars. Friction forces on the flag from the atmosphere mentioned above are proportional to the mass density of the fluid and the square of the speed of the flow. On Mars, the pressure and therefore the mass density of the atmosphere is so low that friction forces on the flag would be almost zero, except with an extremely fast flow, for example during a storm. So it would be pretty much like on the Moon. On the other hand gravity is twice that on the Moon, which would amplify the oscillation phenomenon. All in al, the flag would behave approximately the same way.