Afaict these aren't related.
This is one part of 2000s tech I'm happy to have mostly forgotten about.
You could encode a CD sized video file, burn it, and watch it.
DivX the video codec started out as an unlicensed hacked version of Microsoft’s MPEG-4 v3 codec binary. Since it wasn’t a commercial product and was legally dubious, the author called it DivX ;-) with the smiley in the name.
When it became unexpectedly popular during the dot-com boom time, someone of course set up a DivX company that dropped the smiley, eventually rewrote the codec, and presumably acquired the trademark from the defunct DIVX (or just took it over if the registration expired, I don’t know).
and then, iirc, this is where xvid come into being. I think it was the same codec just re-written and given back to the opensource world, hence the reason for naming it "divx" spelled backwards.