It makes sense, but it's not direct. It means no one has ever given her a wedding ring like in the movies. Lyrics can be hard to understand because they are artistic and subliminal. I wouldn't say they "don't make sense"

"to cut your teeth" on something means to learn from it, usually when you are young or just starting out at something. “When I started to learn Spanish I cut my teeth on Mexican movies so I know a lot of slang.” The lyrics ("Royals" by Lorde) are: I've never seen a diamond in the flesh I cut my teeth on wedding rings in the movies She has never seen a diamond in real life ("in the flesh"), she only knows about them from movies. ie. she is not rich. It's true this is a strange way to say it but it is a song and you can say anything you want in songs. ;-)

Mike, there are lots of lyrics in English songs that don't make sense. The songs written by John Lennon of the Beatles usually do not make sense. If you can explain to me any of the lyrics of the songs written by Stevie Nicks of Fleetwood Mac I will send you $20. For a lot of musicians the priority is to sound well. Communicating anything of value is a low priority.