Why is Singin' in the Rain a musical? Um, did you not see all the singing and dancing? Some of it was in the rain.
All kidding aside, Singin' in the Rain is actually a pretty interesting entry in the pantheon of movie-musicals because all of those songs and epic dances pop up organically. Nobody just bursts into song for no reason; there's always narrative motivation. Many earlier musical films were just stories with musical numbers thrown in—musical revues, essentially, without using the songs to drive the plot.
Take "Make 'Em Laugh," for example; Cosmo launches into that song as a means of telling Don to lighten the heck up. "Moses Supposes" is born out of Don and Cosmo poking fun at Don's diction coach and riffing on one of his ridiculous tongue-twisters. Don sings and dances in the rain because he's in love, and it happens to be raining. These naturally occurring musical numbers even incorporate props that have context, like an umbrella and a streetlamp on a dark and stormy night.
Movie comedies come in more flavors than gelato—everything from sports to slapstick to sci-fi. Your baseline comedy just wants to make you chuckle at the world around you by pointing out the truth about it. Singin' in the Rain pokes fun at the ridiculous nature of the movie industry—specifically how very little in Hollywood is actually how it appears. The biggest, brashest, literally loudest example of this is Lina. On screen, she's elegant, glamorous, and refined. Off screen, she butchers the English language, fights dirty, and sounds like a malfunctioning smoke alarm. The first time you hear her voice, you can't help but giggle.
Singin' in the Rain also wrings laughs from the movie industry's less-than-magical-reality by criticizing it. That makes it a satire, which is a special branch of comedy that uses exaggeration and irony to show how silly or stupid something is, usually by highlighting its shortcomings. Ever read The Onion or binge-watch 30 Rock on Hulu? Those are both classic examples of satire.
Singin' in the Rain isn't all tap dancing and pratfalls. It's also a love story between Don and Kathy that adheres pretty closely to the conventions of the genre. First, Don and Kathy have a classic "meet cute" when he jumps into her car. Then a misunderstanding keeps them apart—we're talking about when Kathy fears the fallout of accidentally thwomping Lina in the kisser with a cake at R.F.'s party. Once Don and Kathy laugh off the misunderstanding, they start falling head-over-heels for each other ASAP. Then villainous Lina butts in and tries to break them apart by threatening to ruin Kathy's career. But ultimately they triumph: Don gets the girl, Kathy gets the guy, and they both get a sweet movie deal with Monumental Pictures. As they say in the movie biz, that's a wrap.