(Assuming you are reading the book in English translation) The narrative character, a French expatriot in Algeria, uses language that endistances himself from his emotional surroundings; that is why the opening lines of Camus’ book are so famous:  “Mother died today.  Or maybe yesterday.” On the surface this remark simply refers to the probable delay of news traveling to a distant land, but by the use of “nitpicking details,” Camus demonstrates the character’s lack of emotional response to news that should have had a devastating effect on him.  Later on in the book, when describing the actual shooting of a stranger on the beach, the narrative character displays the same unemotional distancing and indifference to the crime itself.  Camus uses the linguistic device of understatement as a parallel to the existential concept of freedom from guilt—since there are no predetermined right or wrong human actions.  The Stranger, then, refers not merely to the victim, but to all of us – strangers to a universal set of laws, strangers to each other, as we form our “meaning” by our actions.

As to personal identity, the narrator is clearly acting outside the expected behavior of social beings.  Camus' point is that we do not have an obligation to act inside a pre-determined set of "rules," but actually design ourselves by our choices.  The intriguing non-motive of the character's actions is a strong metaphor for the basic existential tenet: "Choose; that is, invent."  The absence of any "psychological" description of the character is Camus' way of saying there is no such thing as "human nature."  Rather, Mankind is only the accumulated history of individual's choices.  The narrator, by his (to the reader) senseless action, has added his identity to this history.  For a social sciences student, the novel is a strong image of what happens when one steps "outside" previously accepted behavior.  Read also Andre Gide's The Immoralist for another example of how fiction can serve the social scientist by creating fictive "mis-en-scenes" where characters make unique choices.  Fiction is like an experimental laboratory for social sciences.  See Umberto Eco on the function of fictive characters in society.