Status is central to how we meet and regard each of the play's central characters. In the very first scene, we get a little exchange on status between the boatswain and Gonzalo. When the boatswain is troubled by the non-sailors distracting everyone on deck, he says, "Hence! What cares these roarers for the name of king? To cabin: silence! trouble us not." To which Gonzalo replies, "Good, yet remember whom thou hast aboard."
We haven't gone 25 lines into the thing before the play's courtly elements are reminding everyone to be careful of their titles and the status that goes with them. This also sets the tone for the play, in that the natural elements couldn't care less about men's titles – each is equal under the sun. This tension is highlighted in the play as Gonzalo lays out his ideal society. Still, at nearly every turn, some individual is playing up his supposed nobility, stolen nobility, or potential nobility.
Prospero reveals to Miranda after all these years that he's a duke and she a princess, and when we meet Ferdinand, his father's body hasn't touched the bottom of the ocean (for all he knows) before he lets it drop to this pretty girl that he's the new King of Naples. Stefano would be king of the island, but the very real evil behind plotting to murder Prospero seems undone once Stefano's real status is revealed (his identity as the King's drunken butler). Stefano doesn't exactly strike us as the type to be leading the class revolution, and so he's a foolish little man, not a serious traitor, and poses no real threat the crown. Gonzalo fantasizes about having the title of sovereign, and Antonio has stolen nobility while Sebastian is urged to get the same. Even Caliban claims he was master of the island before Prospero stole it from him.
In this pastoral setting, far removed from the court, Shakespeare's characters are still quick to throw around their courtly titles, save perhaps Ariel and Miranda. It's a fine lens to view each character's natural nobility in contrast to their pride in pomp.
Education is also a characterization, even if it's more peripheral than status. Prospero's dukedom was stolen from him while he pursued his learning in the liberal arts, a seemingly more noble pursuit than power politics. He says Miranda has been elevated by her education, and Caliban is held up as a contrast because he too learned Prospero's language and science. While he once loved Prospero for it, its greatest gift to Caliban has been to curse, either because of his ignoble nature, or because learning can only further the sense of indignity an individual suffers when they are wrongly thrust into servitude. Learning is important for the freedom it either provides or shows to be absent. See, they weren't kidding when they said knowledge is power, even if you are an angry little son of the devil.