I write this answer as someone who reviews films for a living and is also intimately familiar with Britten's music. I'm hard pressed to think of any relationship between the structure of the film's story and the structure of The Young Person's Guide to the Orchestra. The musical piece is structured as a theme and variations, which the spoken narration briefly refers to. Such a work begins with a theme (either an original one or one by another composer, as is the case here) and then is followed by a set of variations, in which the composer alters the original theme in various ways. In The Young Person's Guide, Britten uses the different variations to showcase different instruments in the orchestra. In Britten's Variations on a Theme by Frank Bridge, he turns a strain composed by one of his music teachers into a classical minuet, a Viennese waltz, an Italian opera fanfare, and a somber funeral march, among other things.
For a film to take the structure of a theme and variations, it would have to be very episodic, moving from one character or set of characters to another and having each vignette somehow reflect the main plot. I've seen Moonrise Kingdom several times, and I don't detect this structure at work. The situations of Suzy's parents, Scoutmaster Ward, Captain Sharp, the other Khaki Scouts, etc. don't really mirror Sam and Suzy's romance in any appreciable way. I'm sure there's a film out there that is structured as a theme and variations, but it's not coming to mind at the moment.
If the structure of The Young Person's Guide doesn't reflect the structure of the movie, then what's it doing in the movie? Well, Wes Anderson is apparently a longtime Britten fan who performed in a production of Britten's opera Noye's Fludde when he was 10 or 11. He said that he began the film with a notion that Britten's music should be used to illustrate this story. Most of the other Britten selections he found for the film, but The Young Person's Guide is one of Britten's most popular works. (The composer wrote a great deal of music for children.) It makes sense to begin the film with a piece that's well-known. It's also logical, within the story, that Suzy's parents would have this record in their house for their children to listen to. The music makes a backdrop to the opening credit sequence, in which we're introduced to Suzy and the members of her family. Visually, the sequence does emphasize how each member of her family is occupying a different space in their house, which chimes with the psychic distance between Suzy's mother and father, and between Suzy and her parents. All this to music in which Britten compartmentalizes the different sections of the orchestra. We then hear the music again near the end of the movie, when Suzy is back with her family. Specifically, we hear the end of the piece, when all the instruments in the orchestra once again play together.