As soon as you I read your opening post I wondered how quickly I would find this answer (based off of the grounded equalized setting.)It would have been an interesting option to place V's apartment in a different megabuilding (different neighborhood) for each background origin. They are largely the same, so the apartment could be identical.
It would have given each origin access to different side jobs and gigs in act 1, which would have made the early game experience more varied. (You'd have to design quests with this in mind from the start, of course.)
I personally find this very intriguing if it were the case. I dont even think it would've been that difficult to make happen.