You have to understand the whole-part doctrine in Vedic philosophy. The whole is the bigger idea, and the parts are the details that elaborate that idea. A simple example in Conceiving the Inconceivable is that of mammal and cow. There are three different notions of mammal here.
First, mammal is an idea, that exists when cows, horses, dogs, and cats don’t exist.
Second, when cows, dogs, horses, and cats exist, then, mammal is the set of all these species.
Third, mammal is immanent in each dog, due to which we can say that they are mammals.
Due to the first, the mammal is transcendent to all the species. Due to the second, the mammal is all the species. And due to the third, the mammal is immanent in all the species.
In the same way, in your body, the person is transcendent to all the body parts–due to the first. Then the person is all the body parts due to the second. And the person is inside all body parts due to the third. If all the body parts are replaced, then because the person is transcendent to all the parts, therefore, the person is the same. Due to the second, right now, the person is some body parts put together. And due to the third, the personality of the person is immanent in each body part.
The Ship of Theseus is used in everyday life. For example, if you take a bank loan, and in 7 years, all the body parts are changed, even then you have to repay the loan because the person is transcendent to the body parts. Likewise, a crime committed 10 years ago is still punishable today because the person is transcendent. This is because the soul is transcendent to all the body parts, including the mind. The mind is also considered a part of our consciousness, but it is not the soul.
If your hand was somehow severed from your body, then your identity is not reduced, even though the hand is separated from the body. Likewise, the hand that is severed from the body is still identified as your hand, even though it is severed from the body. This is because there is something in the hand that is you (the immanent person), although the person is transcendent to the hand.
So, there are three notions of identity. The transcendent person, all the body parts, and the person in each body part. The last two keep changing due to birth and death and the first is eternal. The Ship of Theseus problem refers to that transcendent entity that remains unchanged even if the parts of the ship are changed. The name of a person generally refers to that transcendent entity.
This tripartite doctrine is used to explain the nature of the Absolute Truth, which are called Bhagavan, Paramatma, and Brahman. Bhagavan is the transcendent thing; Brahman is set of all the body parts; Paramatma is the immanent thing inside each of the body parts. If you read Conceiving the Inconceivable carefully, then you will get all these theories as they discussed in detail.