I am not sure it means spiritually most of the time (or at all...even those dead spiritually have a soul)
But I have never looked up the times it is used. I can only think of Ezekiel 28 off hand ("the soul that sinneth shall die") but that was physical death (it was about Israel holding sins of the fathers against their children, God describes His ways a paragraph later).
That said, it can get confusing as having "spiritual life" assumes one already has a soul. But in some instances people believe them to be synonyms.
I like keeping it simple (although most would disagree

but it is really simple).
"It is appointed man once to die and then the judgment".
"Then Death and Hades were thrown into the lake of fire. This is the second death, the lake of fire. And if anyone’s name was not found written in the book of life, he was thrown into the lake of fire."
I am not sure that this "second death" is the death of the soul. It certainly is not dying spiritually as if they possess spiritual life they will not experience this second death.
The reason I say that it may not be the death of the soul as I understand the soul to be who we are (as opposed to our body). I lean towards the wicked experiencing an eternal state as the second death. But I also realize annihilation is a legitimate interpretation and would mean the soul dies.
So I can see God destroying the soul, but this soul continuing to exist in that horrible state.
I agree that those who do not believe will die in their sins. The condemnation is they reject the Light. But they reject the Light because of their sins (their minds are set on the flesh, which is death. They love their sins so they reject Christ).
And I agree that in Christ we do not remain in our sins. But I believe this is because God forgives us based on "repentance", "a new heart", " turning from wickedness", "dying to sin", "turning to Him", etc.
My question was more about Jesus bearing our sins.
So many automatically reason that if Jesus bears our sins they had to be taken away from us snd put on Him.
But these same people do not believe that us, bearing Christ's righteousness, means His righteousness was taken away from Him and put on us.
I do not get the inconsistently in reasoning.
If we bear Jesus' righteousness and He remains righteousness (something we all believe) then why woukd it also not apply to Jesus bearing our sins? This seems to work better with the idea of God reconciling God to man, Jesus becoming like us, Jesus sharing our sickness, etc.
Just curious.