I think it is a TV prop.
It's a great looking prop, very futuristic, very science fiction.
But still a prop. A gimmick.
There is already 3 dimensional chess: ping-pong, basketball, volleyball.
Top Gun.
Fighter pilots already fight in 3D space, since WW1.
The model for dog-fighting is volleyball. It's in the movie. The team work in volleyball is the team work in dog-fighting.

Gentlemen (and maybe women, I don’t really know who any of you are), the discussion of these posts have changed from the original intention of my post. These may be great chess variants, but I want to show chess.com how excited people would be for Star Trek’s 3D chess.
After looking it up, I have found some sets of rules that other trekkies have pieced together from television footage:
This is really detailed, https://www.chessvariants.com/3d.dir/startrek.html
Also, https://meder.spacechess.org/3dschach/chess3d.htm
So look people, I don’t really think that this idea can qualify as some pipe dream, that is not worthy of a proper discussion with chess.com.
If chess.com is reading this, look at the links, see that it is a real game, and please consider it for your website. Trekkies, please comment with support. Other people, please comment with questions about this variant.
I wish you all a great day, and to boldly help us add 3D chess to chess.com.