Now the usual chromatic aberration patterns are purple/green and red/cyan but is green and red possible through simply shooting the photo?
What photo? The video linked in the question is computer generated video. Each frame is computer generated, not a photo.
If you wish to see green and red CA in a photo it's fairly easy to do. Just use negative film under conditions (lens with uncorrected or poorly corrected CA, direction of lights source(s), etc.) that produce CA. and don't reverse the colors when printing/scanning. Green is opposite magenta on the color wheel. Red is opposite cyan. Of course everything else in the frame will also have the color reversed as well.