Overall, what does this movie say about having (or not) an enduring self?

This is an identity question. Because Leonard cannot remember what has happened in the short term, not only does he lose grasp of how time works, but he also gets out of step with his own identity. Unable to accumulate experiences, he becomes a kind of neutral identity, someone who could be anyone at any moment because he doesn't remember who he was just moments before. At one point, he switches clothes with Jimmy Grantz and takes Jimmy Grantz's car—but later he has no way of knowing that he has done so. This begs the question: Who is Leonard and is his memory reliable? While Leonard's memory loss is a literal condition that was brought on by a traumatic brain injury, it is also a metaphor for the experience of being post-traumatic. Leonard is caught in the memory of his own trauma, unable to move forward, frozen in time. Leonard must constantly contend with the fact that he will inevitably lose all the information and memories that he has accumulated since his wife's death and his injury.