Yes, more or less.

It's a great phrase, for that character. It gives you the feeling that he was thinking slightly in the fantasy world (like inside of a shooter game), but suddenly it all became very real (game over). It also serves a good touchstone for the audience to think that this turn of phrase would survive into the era of regular spaceflight.

Although PC based video games may have used the phrase, most people in 1986 would have also known the phrase from coin operated video games. Pacman (1980) prominently announced "Game Over", as did Asteroids(1979). Older games such as Pong (1972) and Space Invaders(1978) did not say "game over", and went with a simpler "Play" sign between games, perhaps to save on memory.

But the Game Over sign goes back further into the pinball games. Games had started using flippers in 1947, but that still left the flippers energized and people could come by and bang on them. By the late fifties, makers added a relay to detect when the game was over, and would turn off the bumpers and flippers. To make sure that no one thought that the game was broken they included a light in the backglass that said "Game Over" or "Match". https://www.ipdb.org/machine.cgi?id=7