Because the nanites are exploiting the brain's neuroplasticity in order to essentially reprogram the relic user's memories and personality. Your neural network changes all the time as a result of lived experiences, strengthening existing neural connections relevant to those experiences, weakening neural connections that aren't relevant, and creating new connections for new experiences. This is neuroplasticity.
What the Relic does is to hijack this process by artificially restructuring the neural connections via nanites to form what is essentially artificial memories the brain never actually experienced. The brain can't tell the difference, since it has to reference its own memory in order to determine whether or not its memory is accurate.
First of all, an engram is not an AI. An AI or personality construct can incorporate engramatic data in its own code to simulate the person the engram is based on (which is why the Alt AI still identifies as Alt and has what can be considered a personality), but the engram itself has no consciousness, it's just a digitized neural map.
As Dr Packard explains in River's personal quest:
"Well, it's designed to store a person's neural engram. Er, sorry... just imagine copying over a person's memories and personality."
Secondly, you say "make room for", yet you refuse to explain how an organic brain can house digital data. The biochip isn't turning the brain into a computer, it's just exploiting neuroplasticity to remove old memories and create new, fake ones based on memories stored in the engram.
Neither of them describe what you're talking about.
No, she says she will inject V's engram (aka a digital copy of their memories and personality) back into their body. She's being vague about how this is supposed to happen, but since engramatic data can't be written directly into the brain (given that the brain is a biological organ and not a digital storage device), realistically the only way to achieve this is by replacing Johnny's engram on the relic with V's engram and then have the Relic's nanites reconfigure V's neural network with their own memories and personality stored in their engram.
The end result is that V - the original, biological V - wakes up outside of Mikoshi with their original memories and personality, almost as if nothing's changed.
I've already explained why engrams aren't AIs.
And the concept of a soul in the cyberpunk genre has nothing to do with the metaphysical or religious concept of a soul, it's another term for humanity. Alt believes the process will eliminate V's humanity because that's what happened to her. She's also spent so long in cyberspace that she no longer has any concept of what it's like to be a biological human being, so nothing she says about the human condition should be taken as gospel.
Yeah this is all BS because neural connections are not genetic - if they were, V's body would have rejected Johnny's engram. The DNA factor was some nonsense introduced to enforce the game's binary ending choices.
On the contrary, not only is this your interpretation, it's an interpretation you've made with zero evidence to back it up.