When you first go to your garage and use the elevator the TV show is actually discussing the relic, I haven't had this repeat. It's with a woman who showcases cybernetics and an actual priest. These two debate what the relic actually is. The woman saying it's immortality, the priest says that it's damnation and just the show of the divide between the rich and the poor - he's right here since only the very rich can actually buy this immortality.
The host steps in saying that the chip is just doing the very same on what the church provides. You give them money, you worship your entire life, you play by their rules and eventually, you might go to 'heaven' with your soul for an afterlife and maybe a rebirth. With the chip, however, the middleman is cut out, you can be reborn again if you slot the chip into a corpse.
It can be argued that the reason Soul Killer is called Soul Killer is that the act of using it is barbaric in regards to most humans understanding - most of them being religious. It kills but also creates life. It can repeatedly have a person live multiple lives without the aid of any god only science. Now you have no fear of what you're doing, because why should it matter? You're not going to hell. You've been taken away from the whole cycle of morality.
We today are people who throw around the meaning of a soul all the time. When someone is evil and without compassion, we call them 'soulless bastards' for example. A soul is just heavily tied to humanity and it seems that it's humanity alone. You don't see many people pondering if other animals have souls. And the only time we tend to on the fly is when we think about animal ghosts, but again, that's a ghost - we never go, oh... animals must have souls, all that often.
If we go back to the basics of what most people tend to think every day terms of 'what is a soul' - it's that you have human qualities. Love, compassion and humanity. Again, it's why when you describe someone as soulless you're saying they're lacking these traits - humanity being the big one.
Having your very being trapped onto a chip, sets you apart from humanity. Why? Again, science here. Love and empathy come from your body being able to work. Endorphins and hormones. You slot your chip out and away from these things you are physically incapable of feeling them anymore. Love isn't some sort of thing that comes from the soul, it's all about science - humans are just romantic and like to think it's the soul. But seriously, love, lust, empathy - hormones. Johnny is only feeling these things now because he's in your body and as a result is getting all the biofeedback - the hormones.
Finally, it brings about the question of how your soul is now owned it is property. It's like slavery but worse. As a black individual and learning about my ancestors, when we were enslaved it was in the body mostly. Slaves did everything they could to keep a part of their souls away from this. You could own the body but you couldn't own their souls, a clear example of this is in the religion of Voodoo, where slaves would continue to worship and adapted it.
Blah blah: V is V as long as V keeps their humanity intact. The fact that V gets their engram put straight back into their body, also keeps their humanity intact and the ability to feel.
Tldr: Humanity is the soul. The soul is your humanity.
Anyway, I rambled long enough.