In the opening of the play, Algy, most likely out of pride for this 'invalid' construct that he has created, constantly repeats Bunbury and Bunburying. At that moment, the reader is relatively ignorant, so the repetition of the term can be frustrating, a frustration that is mirrored in Jack's feelings towards Algy.
Presumably as a critique of the fashion in which British aristocrats readily augmented their personality in order to portray themselves in the best light possible, Wilde has written Jack and Algernon to literally live dual lives. While this may seem like unrelatable, aristocratic grandeur, the idea that people shift their behavior to accommodate circumstance is very applicable in daily life.
Before the gambit ends with Jack's proposal to Gwendolyn, it is left rather ambiguous which 'half' is dominant, Jack or Ernest. Is Jack the Dr. Jekyll to Ernest's Mr. Hyde, or are those roles reversed? However, as the play progresses and Jack is noticeably frustrated by Gwendolyn's love for the name Ernest, it becomes evident that Jack is the dominant half.
Revealing himself as Jack to Gwendolyn would help Jack tremendously as it would help avoid so much of the conflict that later occurs with Algy. However, a rural life would not go over well with Lady Augusta, and after all, it is a story. Contrivances and coincidences must occur.