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.
I'm fairly sure that the very first time you talk to him, he will automatically play something. However, I can't completely confirm this at the moment. The following method may work the first time, but as I've already spoken to him and had my first encounter, I can't try it.
What I can confirm is this:
In order to request music from K.K., you need to back out of the first conversation with him. When he asks if you want him to play the first time you talk to him in an evening, decline, or else he will automatically pick a song to play at random. I tried this tonight because I didn't want to be given another random song like last week, and it worked.
When you talk to him a subsequent time, he will give you 3 options:
- Request a song ("I know a song!")
- He'll suggest one based on a certain mood ("Here's my mood...")
- He'll choose randomly ("I'll let you pick")
The last option you'll have is to back out, which is "Never mind." You can keep on requesting songs for the duration of his stay on Saturdays, but you'll only receive a copy of the song for the first song of the night.
Best Answer
The songs that appear in the Nookings' store are randomized completely. Even if they may seem to be a bit consecutive, all of the songs are randomized. If you're pretty desperate to get a certain song, just keep checking back. Your best bet is to wait for Saturday to roll around.