In Japanese, モブ (short for モブキャラクター) refers to a character who plays a minor role in anime/manga/games - source. The same source claims that the etymology is 和製英語 and traces its origin to Japanese animators in the 1970's.
In English, a "mob" refers to a type of non-player character in a computer game. According to this source the term was invented by Richard Bartle as an abbreviation of "mobile object". I can't find a date for it, but Bartle was born in 1960, so I assume the term was coined some number of years after his birth.
The two words have very similar meanings in both languages but apparently two completely different etymologies. Is anyone able to come up with additional sources that confirm the different etymologies or sources that can refute one of the two proposed etymologies?