I would assume that it will be assigned to the commander.
So CMNDR PS4 might own a red paint job, but CMNDR PC can't use it. If CMNDR PS4 clears save, then his replacement has a red paint job.
Isn't that how it works now?
It does work like that now, but as PS4 uses Frontier Points & PC doesn't, that's to be expected. I'm not asking if paint jobs can be shared though, just if the Arx is linked to the Frontier account or the Platform. I've got loads of credit on my PS4 account & don't tend to play it anymore, so I'd buy Arx to use on my PC Account if it was transferrable.
I hope that clarifies things.