Becoming an immortal is a three stage process for the player character:

1. Born as a mortal, but also as a paragon of the Heart of Nirn - as suggested by Doctor Rhythandius in Chaotic Creatia: The Azure Plasm (this book is linked above - read it if you want to understand this).
2. Killed by Mannimarco. Physical remains utterly consumed. Soul trapped and then subsumed by Molag Bal. Their last remaining echo turned into a daedric-style vestige, and then dipped into the azure plasm to create a soul shriven body. But it's still a "paragon" vestige with some super powers, including the ability to form a body immediately.
3. Absorbs a Skyshard. Re-attunes with Nirn (and with the Spirit of Nirn). Effectively ceases to be a soul shriven with a vestige. The Aetherial magicka of the skyshard gives the player character a divine Anuic focus that acts as a substitute for their lost mortal soul. Thereby becomes an immortal agent of The Divine...

The player character's name "The Vestige" is accurate when the Prophet first uses it in the Wailing Prison, but once they have absorbed that first skyshard it is a complete misnomer. I think this misnomer is a characteristic piece of misdirection that points a finger directly at the Divine whom the player character serves - Lorkhan the "Missing" god. It also helps players make up their own mind about what's going on (or not).

Sure, the Vestige starts out in their cell in the Wailing Prison as a soul shriven vestige that has been dipped in the azure plasm of Coldharbour, but that is only relevant up to the point when the Prophet says:



and then this happens:



Ecstasy. But no Epiphany you'll notice (more misdirection by the Missing God of Creation, and more player choice).

The Prophet's important words here are "Aetherial magicka", "essence of Nirn" and "Lorkhan". Of course, players are free to interpret these words as they wish, or to choose to ignore them, or to completely fail to grasp their significance, but for me they can only add up to one thing - from that point onwards the player character is entirely an agent of The Divine. For me the reference to Lorkhan, aka the Spirit of Nirn, means that the Vestige is in fact a Shezzarine, like Pelinal Whitestrake, Ysmir and eventually Talos.

Wayshrines also point to the player character's special relationship with the Divine. The primary function of Wayshrines on Tamriel is to act as shrines to the divine in one guise or another, as places of worship, not as respawn points for "dead" player characters. The player character is also able to port to wayshrines. In TES III this is called "Divine Intervention".

There is now some very clear in-game corroboration that the player character is not a soul shriven vestige of Coldharbour:
Elsweyr spoiler:
The return of the Vestige's mortal soul is shown in-game to be irrelevant to the Vestige's status as an immortal. It makes no difference whatsoever to the Vestige's experience of "death". It is evident that the player character's original soul is completely eclipsed by the far-stronger divine Anuic focus gained from skyshards.
I have no doubt that the Vestige is an immortal agent of the Divine, but of course, for your characters the choice is entirely yours.