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First, I am not a philosopher, just a curious person (perhaps one could say that is a philosopher :))

If you take Theseus' ship and use all of the wood to build a house. Let's call it, Theseus' House.

Is the house still Theseus' ship, or is it Theseus' house? Is it both? Or is it just wood to which we've assigned a label?

In topology a donut and a cup are the same thing.

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    Even in topology a donut and a cup are not literally the same thing, they are just homeomorphic. One represents them distinctly and then constructs a homeomorphism, a map between two different things. Answers about the ship of Theseus depend on one's theory of identity, and there are many of them. In part, what is or is not considered "the same" is just a matter of convention. For general information see SEP, Identity Over Time. Questions here are expected to be more specific. – Conifold Mar 9 at 9:49
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I feel your intuition partly that there is a question here. We can see in everyday life that repurposed materials used in objects unlike the original object from which it was gathered are given a status of "being from somewhere else" which gives a sense that these objects retain a property of their former usage. In many cases, this is actually an important property of the new object, like if I made a reusable cup from recycled water bottles, I might make it a point of connecting the materials of the cup with their former usage. However, I think that such a case does not create any metaphysical problems like it does in the original Ship of Theseus thought experiment. Houses and ships are of different forms, perform different functions, have different qualities and so they are not close comparisons. One cannot call a house built on land a ship. One might wonder if the house built on land is in the form of a ship. Would that be close enough to a ship to be called Theseus' Ship? My intuition says yes, but I have no clue. Lmk what you think!

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We have concepts such as "ship of theseus" or "house of theseus" that name actual things in the world. The concept corresponds to the thing in the world. The concept is defined in terms of the thing in the world. But we do not have complete definitions for most words we use to talk about the world; our concepts break down at the borders.

So, there is a "ship of theseus" in the world, and we define "the presence of a ship of theseus in the world," to be a condition whereby part of the world is arranged in a certain way that matches our (fuzzy, only-partially-defined) concept. This is what it means for there to be a ship of theseus in the world.

Note that "the presence of a ship of theseus in the world" is different from "the concept of the presence of a ship of theseus in the world"; the first one depends on conditions in the world, and the second is abstract or in the mind. In the ship of theseus problem, "the presence of a ship of theseus in the world" is poorly defined because "the concept of the presence of a ship of theseus in the world" is poorly defined, but they are still not the same thing.

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Philosophy is like engineering, not science, for valid deep meaningful philosophical questions there's usually no definite answer which can be scientifically proved, in addition each word's definition can weigh a lot for one's final conclusion.

In your case, under mereological nihilism, Theseus' ship and Theseus' house are both composites with same "simples" so they both don't exist at all except as pure conceptual names with respect to some different arrangements (relations) of same underlying simples, thus seems meaningless to compare their identity. Under functionalism, probably they have to be totally different. Under Platonic idealism or structuralism they may be interpreted as same or near same since they share certain ideal forms (clearly not same topology but same Hellenistic cultural form since they're both from Theseus) like your donut and cup analogy. Under principle of identity of indiscernibles, since they're still discernible by you, they should not be identical for you...

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