Orsino is the Duke of Illyria, and Olivia is a countess. Orsino is attempting to court Olivia, but she is not showing much interest. Rather, she tells her suitors that she is mourning for her brother who has recently passed away, and she will not marry for seven years. Orsino and Olivia are both very egotistical. Neither character is concerned with the world around them as they are too focused on the drama in their own lives. Orsino's ego is fed through his new Page, Cesario (Viola disguised as a man), and Olivia pines for the love of the same man (Cesario). Both Olivia and Orsino have grave misconceptions of reality and what true emotion really is. They are extremely superficial and change very little throughout the course of the play. It is obvious, however, that Olivia and Orsino are female and male representations of the same person.
Orsino is the duke of Illyria, who is also known as the Count, which Shakespeare had given to him the most complex and entertaining poetical lines of Twelfth Night, making him blurt out philosophical statements in lyrical forms, He loves Olivia, the countess, who have rejected him countless times, but had fallen in love with Cesario (who is actually Viola in disguise), the messenger of Orsino love-sick message, captivated by his boldness and courage to stand up to his purposed task, making him very intriguing.
Orsino and Olivia are similar in their personalities. They are both buffeted by strong emotions, but on the other hand, they unknowingly seem to be individuals who really likes to indulge into melodrama and self-involvement more than anything else. They are very egotistical, showing no particular interest in outside matters, relating to the external environment, and always locking themselves up in their sorrowful world of setbacks and difficulties and stay at home alone, showing that they are very vulnerable within, their heart very fragile to take in any sudden impact.
In conclusion, I think that that Olivia and Orsino are just male and female character and versions of the same particular person