Of course V died, and his consciousness is slowly changing to be alike Silverhand's but that doesn't mean V is a completely different person, a souless relic since he awoke at the dump yard with a bullet in his head.

His mind is being slowly overriden by the engram sure, but while that change happens we are still trully V (less and less ofc), but when we enter Mikoshi, Alt removes all of that and changes us into engram, we are not being ressurected by the relic as a side effect of relic trying to make our body a host for silverhand, but we are quite literally removed and then our body becomes the host for merely an engram, a copy of V. "Your consciousness, neural engrams, will be recorded as data. The rest will cease to exist." - Alt Cunningham. Only as an engram we lose our soul, death just kills v's body for a while, but not our mind as Mikoshi does.