Why does the speed of commercial airliners fluctuate, sometimes as high as 1,060 km/h or as low as 800 km/h?
Information displayed to passengers through the entertainment system often gives the aircraft ground speed rather than airspeed.

Wind affects the ground speed, you're right about that.

Few things out of the way first.


Pilots do not use ground speed for flying, instead they use indicated airspeed.
Mach number is not derived from the ground speed, as it is dependent on air temperature, instead it is derived from the true airspeed. Notice we already have two kinds of airspeed. There are more still.
Aircraft type (for example 737 vs. 777) determines "top speed". Not all airliners are the same. Those two types range from Mach 0.77 to Mach 0.85, roughly.


In less words than Wikipedia which I linked:


Indicated airspeed tells the pilot the aerodynamically relevant speed to be used for flying.
True airspeed is the actual speed relative to the surrounding atmosphere.
From the true airspeed, add or subtract tail or headwind, and you get ground speed from that. Typical wind experienced by the regular subsonic airliner is fast (+100 km/h). The tail or headwind component would vary depending on the plane's direction and the wind's direction.
Finally, Mach number is a function of temperature and true airspeed (notice, not ground speed). The higher you are, the colder it is. Also very important, Mach number is a ratio, never a speed. Since the speed of sound varies as the atmosphere changes density and temperature at the different altitudes.


Here's a nice story, a 1200 km/h subsonic flight.

Summary:

Pilots don't use ground speed (for flying), mach number is not a speed.