Comedy, Drama, Parable
By definition, a traditional comedy doesn't have to be super silly. But this comedy definitely is.
The Miser has one of the most traditional markers of comedy: a happy ending. The evil villain is overcome and most everyone can finally chillax and be happy. Most traditional comedies end with a marriage of some kind (see Shakespeare's Measure for Measure for a good example). And this play ends with not one but two weddings. Double your pleasure, double your fun, y'all.
The other hallmark of a comedy is that it's entertaining. And The Miser is entertaining like whoa. It panders to every audience desire, including getting to hate a really hateable dude, getting to see pretty people end up together, and every getting talked to by members of the cast.
The Miser is also, of course, a drama in that it's a work produced for the stage. It doesn't have to be dramatic (kneeling down in the rain, hail of bullets, tears a'plenty) to be drama. It just has to be a work that is produced with the sole intention of having people in costume doing their thing up onstage while an audience relaxes in front of them. It also emphasizes the stage itself—the flat wooden thing on which the action occurs—by making important plot points revolve around who is on or off of it.
In The Miser it's important that Valère, for example, is offstage when Jacques tells Harpagon that he stole the cashbox. He's offstage and is therefore out of earshot.
Finally, The Miser is a parable. Even though the situation in The Miser is pretty complex what with the double weddings and the theft intrigue, it can be boiled down to a hard-hitting life lesson: don't be a stingy d-bag. If you are a stingy d-bag, people will hate you. So because you can walk away from the theater after seeing The Miser with the warm, fuzzy feeling that comes from having learned how to live well, it gets the parable stamp of approval.