Any planet or moon with a sufficient axial tilt that rotates with respect to its star significantly more frequently than it orbits the star (no tidal locking between the two) will experience the same phenomenon (a nighttime lasting more than one rotation period).
The polar shortest day is not totally dark everywhere inside the polar circle, but only in places within about 5.5° of the poles, and only when the moon is well below the horizon. Regions located at the inner border of the polar circles experience polar twilight instead of polar night. In fact, polar regions typically get more twilight throughout the year than equatorial regions.
For regions inside the polar circles, the maximum lengths of the time that the Sun is completely below the horizon varies from zero a quarter degree beyond the Arctic Circle and Antarctic Circle to 179 days at the Poles. However, not all this time is classified as polar night since sunlight may be visible because of refraction. Also, the time when the Sun is above the horizon at the poles is 186 days. The numerical asymmetry occurs because the time when the Sun is partially above the horizon is counted towards the daytime. Also, the above numbers are average numbers: the ellipticity of the Earth's orbit makes the South Pole receive a week more of Sun-below-horizon than the North Pole (see equinox).
As there are various kinds of twilight, there also exist various kinds of polar night. Each kind of polar night is defined as when it's darker than the corresponding kind of twilight. The descriptions below are based on relatively clear skies, so the sky will be darker in the presence of dense clouds.
Polar twilight occurs in areas that are located at the inner border of the polar circles, where the Sun will be on or below the horizon all day on the winter solstice. There is then no true daylight at the solar culmination, only civil twilight. This means that the Sun is below the horizon, but by less than 6°. During civil twilight, there may still be enough light for normal outdoor activities because of light scattering by the upper atmosphere and refraction. Street lamps may remain on and a person looking at a window from within a brightly lit room may see their reflection even at noon, as the level of outdoor illuminance will be below that of many illuminated indoor spaces.