The replaced parts become a part of the old one. Thus when the last original part is replaced, something added later has already become part of the original, so there is no new ship, just one ship with varying ages of it's parts. You see the object as the whole, for the purpose for which it was intended, not as the sum of it's parts. Otherwise we'd be able to murder amputees without repercussions.
Does a replacement part not have an origin? While I would agree that objects make up a whole with a greater purpose, is it not also reasonable to assume that what gives an object its purpose are the objects which make it? If I put a loud replacement motor on a fishing boat then what has it become? Certainly no longer a fishing boat, since there would be no fish around to catch.
Furthermore, I wouldn't call amputees objects.
But, if you do want to go there, then wouldn't you agree that changing one part of a human can drastically alter his or her identity? What would happen if a musician was denied the use of the body part he used to play his instrument or his voice? Of course he can still write music, but the difference between composers and performers is also immense. What if a healthy athlete were given the lungs of a chronic smoker? Could he truly practice his passion to the fullest or would he have to fundamentally change his identity, his fundamental purpose, to continue living purposely in the world? I would say yes.
Most of what I have mentioned are repercussions of replacing *one* part on a whole, all of which are significant. If we are assuming that identity originates in the purpose of the object, then surely replacing *every* part of an object would make it an entirely new one.