Everyone likes Mary Poppins. Everyone. Even Lin-Manuel Miranda. 

Ghostface Killah from The Wu Tang Clan. The writers of the Simpsons. Vast oceans of kindergartners, and vaster oceans of nostalgic adults. They all love this movie. It has no enemies. (Well, except for P.L. Travers).

Mary Poppins still pops up in pop culture references—that's three pops in one sentence. Such is its fandom.

Ghostface Killah references "Supercalifragilisticexpialidocious" in a song entitled "Buck 50" (which, unlike Mary Poppins, is not family friendly), and The Simpsons parodied the movie in an episode where a nanny named "Sherry Bobbins" comes to look after Bart and Lisa. Instead of helping them clean up Bart's room through magic, Sherry Bobbins teaches them how to clean the room by randomly throwing things under the bed, explaining "It's the American Way!" 

Also, the fact that Disney made a movie about how Mary Poppins was made, indicates just how famous it is. Saving Mr. Banks is all about how Mary Poppins came into existence, detailing the fight between the character's creator, P.L. Travers, and pretty much everyone who worked at Walt Disney. The fact that big name stars like Tom Hanks and Emma Thompson were cast in the movie demonstrates how much Disney thinks people might want to see this—it's not a Lifetime movie starring a bunch of no-names. (Source)

But Mary Poppins isn't popular because of the tense backstage clashes that attended its making. No. It's popular because it's got great tunes, it's lighthearted and fun, and there are no sudden and unexpected deaths (like, say, in Bambi—we're still not over that one).

People need Mary Poppins' "spoonful of sugar"—to help the bitter medicine of everyday life go down.