It's the whole star trek teleporter thing where you're always a copy, and the previous copy dies.
Actually is another plothole. It is explained that the conscience part remains, when you go to Mikoshi, you connect and wake up in the elvator, meaning your conscience uploading to the system.
What isn't sure is the soul part, if an enagram is still a person, or just data, etc. You can argue that with Johnny and Alt.
BUT
Then Arasaka ending goes brrrr:
Saburo is dead, but there is a copy in Mikoshi, so they didn't killed him when the enagram was made, so the whole soulkiller thing doesn't make sense.
Shards along Arasaka basement tell that the enagram thing doesn't works always at 100%, can fail, also that there are many copies of a single person depending of how much they pay.
Lizzy Wizzy quest explains that her manager got her scanned and there is an enagram of her in Secure Your Soul, but she is still alive.
And regarding of the chrome pop-star, she sucided in stage, and Trauma Team manage to revive her with full boy replacement with cyberware, why can't that be done to V?
So in the same game, in the same universe, the conscience goes with the enagram or dissapears depending of the writer needs.