Being the same person is based on our consciousness I would think, not the various cells that get replaced. We essentially are always the same person unless a disease takes away memories that make the person who he/she is.

well if you think about it in terms of the law and registration and all, Its registered as the same car, just with modifications and removed parts. So same car in that case

Maybe.
If you replace the parts with ones that are the same, I guess it's the same car.
If you take a lambo, and replace it with parts from a Toyota then it's not really the same.
It looks the same, but its different? I guess it all depends on how you look at things.

To clear this up a little more, watch the movie Robocop (2014 version - don't get me wrong the classic is great but the newer version makes the discussion easier to understand)
Robocop is about a police officer named Alex Murphy who gets critically injured by a car-bomb. He is selected for the 'Omnicorp program' and he undergoes a procedure where is he is 'enhanced' with a cybernetic body and software.
After the procedure, Alex is shown that only his heart, lungs, throat, head and right hand are left of his original body; the rest is robotic. His brain is now controllable by use of software (ie. his chemistry levels/emotions can be altered by computers).
Coming back to the question: "If a car gets all of its parts replaced, is it the same car?", it depends on the sentimental value of the thing/object you're talking about.
Knowing Alex Murphy before the incident, you have some sort of emotional attachment (can be a tiny amount, can be a large amount). Even after the incident, where you see him having only a time amount of him left, you still recognize him as the same person but with 'upgrades', regardless if he still IS actually Alex Murphy or just a machine, instead of a completely different person/thing because there is more robotic parts than human parts.

Another movie came to mind when reading this thread: Robots.
Basically, the bad robot takes over the good robot's company and discontinues the making of spare parts for robots, only making room for new parts/upgrades. He then pitches the slogan, "Why be you, when you can be new?"
I'm not 100% sure that this fits in with the current topic because it asks "why be you..", where that's about self-confidence and all that shenanigans blah blah blah, but then again, it also brings up the same question as the thread "is something still the same after it undergoes change/upgrade?" so yeah.
This article paints a good picture on the topic, I think: http://www.ncregister.com/daily-news...ou-can-be-new/
(Ignore that religious junk at the bottom of the article)

Let me pose it this way:

If you kicked everyone in clan RAWR, one at a time, replacing them as they are kicked with new players, removing threads from the board and replacing them with new ones, until everybody in RAWR was new and every thread was typical unofficial clan BS- would it be RAWR?

(I'd argue to a lesser extent this happened to RelaxAll)

And I'd say no it's not RAWR.

Technically speaking, it would be the same clan because the clan never died and the name of the clan is the same.

Generally speaking though, it would not be the same clan because the original members are no longer there and therefore the sentimental value and integrity of the original RAWR clan is gone (although some people, if not all, would feel nostalgic about the old clan and its members).

not the brain

Well, I don't think so. Think of a new Porsche with a 1990s engine. Would it be the same? No.

At the point where with all the old parts you could rebuild the old car/ship then the new one is now a new car/ship.

Is how I've always considered this paradox.

By 'new car/ship' do you mean different car/ship?

Let's say you bought an old Toyota car at a used-car dealership. Then you go ahead and change every aspect of the car's looks so now it resembles a Nissan or any other brand of car. You go back to the dealership and say, "Hey I'm looking to sell my Nissan", and end up selling it, is that fair?
Deep down in the back of your mind you're thinking it's the same car you bought in the first place.