If you took a car and replaced all of its parts until none of the original parts are intact anymore, is it the same car or a completely different car

In the UK you would, there's a score of parts you can replace before it needs an inspection and re-registration.
That's the big stuff like frame, suspension, engine and all not oil filters and stuff.

Interesting analogy to this... Is a dead person the same as a live person... Afterall, the original parts are still there?

certain parts are key parts..like the chasis
replace those and yes

It would be a different car, but it would not need to be re-registered here unless those changes put it into a different class - like a car to a light or heavy truck, etc....

Here the VIN tag identify's the viehicle.
As long as that tag is intact ist the same car.

Triggers broom...

The police would stop you for stealing a license plate lol