To answer simply: Vern's Volvo is whichever Volvo Vern chooses is Vern's Volvo.
To answer not as simply: Vern's Volvo is more than the sum of its parts – that is, the fact that his Volvo is no longer made up of any of the original parts has no bearing on whether that Volvo is “Vern's Volvo.” What actually matters is the connection that Vern makes with his car, because without that connection, Vern's Volvo would just be a volvo, not even worth a capital letter.
I believe that identity is an entirely subjective thing – there is no objective essence or identity to an object. Something has meaning because we give it meaning, and that meaning can vary from person to person. My five dollar bill is worth five dollars because it's accepted that it's worth five dollars – if no one accepted the bill as money or acknowledged that it had value, then it's just a piece of paper with a picture of Lincoln on it. For that matter, human life only has value because we want it to; if people believed that human existence has no meaning or worth, then human life has no meaning or worth. So it follows that whichever Volvo Vern chooses is “Vern's Volvo” IS Vern's Volvo, regardless of whether it's Grace's remake or the Volvo that Vern's been driving around. In the same vein, the other Volvo isn't Vern's because he didn't choose it, but if he had chosen it, then it would be the Volvo.
The importance lies in the choice, which can be a conscious decision or not. It's also a choice only Vern can make. I can sit here and put myself in Vern's shoes and think about the choice that I would make if it were Nick's Volvo rather than Vern's, but I can't make the decision for him. This sort of thinking, that the essence and identity of something is subjective, applies to more than just the metaphor of Vern and his Volvo. It applies to existence on a grand scale, and even the nature of truth itself. If one accepts that Vern's Volvo is the pile of scraps (or Grace's eventual remake) because those are the original parts, then it implies there's an objective truth independent of us – Vern's Volvo is Vern's Volvo regardless of what he thinks is his Volvo. But on the other hand, if one accepts that Vern's Volvo is whichever Volvo Vern chooses is his Volvo, then it implies that there's a more subjective nature to truth and identity – we choose what is actually the Volvo, regardless of what parts it's made of, and we choose based on our personal values.
