Red and Blue have been used as juxtaposed colors for a long, long time now across a number of fields including politics, sports, war, and video games. The color scheme seems almost ubiquitous (though it isn't the only color coordination used to indicate conflict).
This likely comes down to color theory and symbolism. Red and Blue are two of the three primary colors, and any combination that uses yellow just doesn't come off as bold. Additionally, red is often used as an aggressive color, and is very obviously a stand-in for blood and death. Blue is often used as an indication of safety or protection.
So, having used red vs blue in a variety of contexts over hundreds of years, naturally the dichotomy made it into video game multiplayer. The Halo series, and its subsequent web series spin-off, have made the Red vs Blue multiplayer tendency in games to be rather iconic of video games as a whole.
But where did it start?
As far as games go, the best known use of "red vs blue" for multiplayer goes back to the ROM days of the 1980s. Speedball, Salamander, Twin Cobra, even Contra all used red and blue to differentiate the first and second players.
However, Atari's Control! just might be the first ever use of the red vs. blue dichotomy in video games. Released in 1977, Control! Has different colored tanks, planes, and jet fighters for each player. Including, you guessed it, red and blue.