The Music Box
The Music Box is actually a family of furniture items, only obtainable by the method you mentioned. Like all other furniture from Cyrus, it is not possible to take one of his products apart into the original materials/furnitures used to make it.
When Cyrus creates the Music Box, the result is a colourful version of the Giant Clam Shell that, whenever opened, will play a little music-box-like version of your song, before closing shut once more.
You can think of the Music Box as a larger, physical, standalone counterpart of the song used to create it, as the Music Box will retain its name when created.
Additional Copies of Music
As for obtaining another copy of the song, if you have access to the Timmy & Tommy Shopping Catalog from upgrading the Nooklings' shop at least once, you'll notice that there is a category in the catalog for music.
Although you cannot order Music Boxes this way, all of the K.K. records you have ever received or will receive in the future will be recorded in the catalog. Each of these records can simply be ordered from the catalog for 3,200 bells.
Therefore, by purchasing your additional copies from the catalog, you are free to request new pieces of music from K.K.Slider each week.
Best Answer
There are two ways to get music:
or
You'll need to get either Club LOL or several renovations to the Nookling store in order to get music for yourself, but if you have a friend who is willing you help you out, they are tradeable, and you can buy them from Nookling stores in other players' towns.