With modern video editing, you could use any color. That bright green color is used because it's quite a bit different from tones in skin and natural tones, and it's easy for equipment to identify the color and get a good separation between the background and the subject.

With older equipment, green had the added benefit of being the "base" color for the video signal (and other colors were created by adjusting red and blue levels). This meant the equipment was particularly sensitive to this color and it was easier to make equipment to lock in on it and treat it as transparent.

Up through the 1980's the same effect was done with film and using the color blue and special filters to make "film sandwiches" that could combine multiple images into one picture. The technique had been used since the 1930's, but it became more sophisticated with each decade.