The slang generated by the real world of cops and robbers is often colorful, but sometimes misleading. A cat burglar, for example, deftly and daringly climbs great heights to get his booty. He’s not a felon who filches felines. But a mugger is not only linguistically related to mug, the slang word for one’s visage, he’s also very much physically in your face.
A mugging is a street crime in which the perpetrator forcefully removes your valuables. He often grabs his victim from behind, putting his hand over the mouth to prevent an outcry, which is how the act got to be a “mugging.”
Mug as a slang word for face originated with old drinking mugs, which were often decorated with distorted caricatures of faces. You might even end up looking that way if you drank too much of what was in the mug.
Source: A Second Browser’s Dictionary (Common Reader Editions) by John Ciardi