The Ship of Theseus is one of the more illuminating thought problems in philosophy, and it having been around for something like 2500 years indicates how insightful the early Greek philosophers were.
What the thought problem shows is that if one takes a reductive approach to identity, where identity is taken as identicality down to the reduced elements of a structure, then NOTHING in our physical universe, except perhaps for some elementary particles, are identical over time. This presents a major problem for reasoning, as classical reasoning is only valid if A ≡ A, and this thought problem shows that A ≢ A.
One major approach to identity taken based on this A ≢ A problem, is to accept that there IS no valid identity in our world. That objects in our world are heaps, composites, without any essence. Therefore identity is only an approximate state, used for pragmatic purposes, and logic, while often giving useful answers, COULD be invalid when applied to these soft-edged "A"s.
Hume was an articulate advocate of extending this "heap" or "bundle" theory of non-essence of things to selfhood. He pointed out that thoughts and experiences changed radically over a short time, and postulated that we are just a bundle of impressions. Selfhood today is generally identified more with memories, and personality inclinations, than with experiences, and these two are more stable than experiences, so Hume's radical variability moment/moment is not widely accepted today. But these other two also both change over time, hence his point that A ≢ A for selfhood is still very true. Hume lived in an era when logic was considered more absolute than it is today. And the Pragmatic alternative, where some ideas that are invalid in absolutist terms might still be "good enough approximations to get by" was still a few centuries off. So Hume followed the absolutist thinking of classical logic about selfhood -- if A ≢ A for selfhood, then in classical terms assuming "self" is a fallacy.
But in a more pragmatic worldview, a selfhood that is not reductively identical moment to moment, and is appreciably different year to year, could still be a useful item to identify and operate with in the world and with reasoning.
There are alternatives to the logical "objects are not real" and the pragmatic "both objects and reasoning are useful but not fully reliable". One of those is available to non-physicalists, where one can postulate an essence to an object, over and above its reductive elements. This essence could be part of an abstract realm such as with Plato's Forms, or could be a feature of consciousness or spirit. An example of the latter is the dualist philosopher Richard Swinburne who postulates that selfhood has an essence, a "thisness" that is maintained over time, despite changes in personality/memory etc. Swinburne's essence is clearly a soul, although he does not describe it as such.
Another alternative is to dispense with the reductionism that Hume and The Ship of Theseus assume is what matters for identity. The philosophy of science today has accepted that there are emergent structures in our universe, and these emergent structures are functionally independent of their reduced components. One can then tie identity to functions that an object performs, rather than to its constituents. Once one does that, there would then still be a Ship of Theseus, and it would be the continually repaired vessel floating in the Thebes harbor. And a set of discard boards, reassembled -- would then NOT be the Ship of Theseus. The function that defined which was THE ship would then be the continuity of the repair crew activities. Per this approach, if the repair crew were to leave, and the ship boards were broken and scattered, then a new crew reassembling and repairing as best they could, would still not be able to make THE Ship of Theseus, as they would never be sure of the provenance of the components and design. But add in pragmatism, and Ship of Theseus V1.1 might be a good enough stand in.
Emergence and functionalism lead to pluralism in science. And pluralism of reference frames — leads to logic explosions. Therefore, a noteworthy outcome of this thought problem is that for all three of the bundle/heap/no-objects, functional/emergence/pluralist, and the pragmatic "identity is approximate" answers, classical logic cannot be relied upon in our world. The only ways that applying logic to things in our world would still be fully valid, would be if they have an essence -- and Plato's Forms and their equivalent are widely rejected today -- with the sole exception of possibly selfhood.
