In Twelfth Night, why does Duke Orsino insist on continuing to call Viola "Cesario" at the end of the play?
A psychological reason for Orsino continuing--temporarily--to address Viola as Cesario may be presented this way. Orsino identifies Cesario by outward appearance, once a new identity is revealed, the appearance will continued to be associated with the former identity: Cesario. Orsino says that "when in other habits you are seen," when she has another appearance, he will address Viola as his "mistress and his fancy's queen" ... and by her woman's name, Viola. This appearance-tied recognition of a new name is a final salute to the theme of allusion versus reality.
Yes, yes, I've read the various interpretations on this subject, but I always tend to think the simplest answer is the right one. In this case, I want to give it a real-life twist. I have a feeling that the reason why Duke Orsino continues to call Viola "Cesario" (even after Orsino has already learned of Viola's true gender) is the exact reason why almost all of my former students insist on calling me "Mrs. Thompson" even looooooong after they have graduated from high school. Some of these students are, literally, only six years younger than me and my current friends on facebook, ... graduated from college, ... finished graduate school, ... many of them parents, ... some of them priests or nuns. Here they are still calling me "Mrs. Thompson." Two words: second nature.
In Twelfth Night, Shakespeare points to the arbitrariness of sexual attration and love. However, he also points to the fickleness of human beings. Like Olivia, Orsino is capricious, too, is his affections. In fact, it seems that he is more in love with being in love than actually loving someone. That he can retain affection for Viola/Cesario once her identity is revealed says more about his appreciation for friendship than his understanding of love.
This is an excellent question, and critics have much debated why it is that in Act V scene 1, Orsino continues to address Viola as "boy" and then as "Cesario" even after Viola has revealed herself to be a woman. Of course, Orsino's actions result in much uncertainty regarding Viola's true identity. Note, for example, how Orsino, at the end of the play seems to acknowledge a dual identity at the heart of Viola's character:
For so you shall be, while you are a man;
But when in other habits you are seen,
Orsino's mistress and his fancy's queen.
Orsino seems to suggest that it is possible for Viola to maintain her two identities, and also seems to express some kind of attachment to the male identity of Cesario. We could argue that this is an affirmation of Cesario as an acceptable identity for Viola. By looking towards the time when Viola will become "Viola," wearing female clothes, and becoming his queen, Orsino is not necessarily saying that these two identities are mutually exclusive. Some critics question whether Orsino is not actually more in love with Cesario than he ever could be with Viola, who is, of course, a stranger to him. Thus this ending seems to point towards the bigger theme of the flexibility of gender and the many different roles that we can all have in life.