The first idea is the Buddhist idea of “no self.” The term used by Buddhist is called “anatta” which means “not self” in Sanskrit. Buddhist teachings refer to self as something that is not required. You need to get rid of it in order to develop and improve. This doctrine may look materialist for now but it is important to know the difference between soul and self for Buddha. They are not the same thing. Buddhists believe in a non-physical element that lives on after the body and becomes a part of another body. But self is something else for them. Self is like the characteristics of the body and the non-physical element together. It is the consciousness what makes something what it is. This, for Buddhists, is nothing. It dissolves sooner or later and has no significance.
Is consciousness self or is memory self? John Locke thought consciousness was self but does that mean I am conscious when I am asleep or does my soul disappear when I am asleep and unconscious. I will define consciousness as a state in which my sense organs are awake and perceiving experience or sense data from wherever possible; it is the subjective awareness. If this is the conscious state then there are many times in one’s life when he is not himself because he is not conscious.
And then there was Descartes with his absurd theory of something non-physical interacting with the body to cause movement. We could use Okham’s razor and cut the complex parts of this theory but it is important to know that Descartes believed that anything immaterial cannot be perceived by sense. Reason helps us in knowing there is immaterial mind or soul.
On the question of where in your body a soul can reside, the answer is simplest. That is barely a problem because we have quite an important example from the sciences, the centre of gravity. This, for an object, does not have to be inside the object. It may be somewhere outside the object but it still is and physics has quite a regard for it.
Speaking of identity, if there is no self and it is hard to perceive an immaterial soul then the body is our last resort for identity. However, the physical body is temporal which makes it not so important for identity. The answer to the question can come out from the main problem that we are dealing here. The problem is “everything is changing” and the solution is that even though everything is changing, it is the same too. Somehow, a mosque that changes over time, wears-off like buildings do and is constructed again, or even designed again in a new way for that matter, has something similar to the initial mosque. That something has remained consistent over time. In this ever-changing world there is immense consistency if we look closely and consistency is enough to maintain identity.