Shakespeare's Coriolanus is a classical tragedy in every sense of the word. It is a play set in ancient Rome (hence the setting is classical), and it obeys the rules of tragedy laid out by Aristotle in the Poetics, making it a classical tragedy in form as well as content. Indeed, its simple, linear plot conforms to the Aristotelian unities of time, place, and action more fully than the great Shakespearean tragedies it resembles.
Coriolanus, like Othello and Macbeth, is a great general. His tragic flaw may be ambition, like Macbeth's, but it might be better characterized as pride or arrogance, a contempt for the common people, and a scornful refusal to allow them to determine his fate, which ultimately leads him to fight against Rome. If we accept this reading, then the hubris and hamartia of classical Aristotelian tragedy are the same thing in the case of Coriolanus.
Coriolanus is arguably the most classical of Shakespeare's tragedies. It is seldom mentioned alongside the great Shakespearean tragedies (defined by A. C. Bradley in his influential study, Shakespearean Tragedy, as Macbeth, King Lear, Hamlet and Othello), but Aristotle would certainly have regarded it as much closer to the classical paradigm of tragedy than Macbeth, principally because Coriolanus is a more noble character. Indeed, it may be that the closeness of Coriolanus to the classical model is one of the chief factors that prevents it from rising to the emotional and verbal heights of the great Shakespearean tragedies. The cold, marmoreal Coriolanus is too close to Aristotle's perfect hero to interest the audience.