Does this mean "childish music" or the music that belongs to the children (like a CD) ?
I think it'd be the former. I'm not sure what the latter would be exactly, but I think that would be "the music of the children," with children being treated as a noun in genitive case. (Музыка детей?)
"kid's music" as opposed to "kids' music" was marked as incorrect. Now, that is a true ruling, but this is a minor point of English grammar (i.e., the correct placement of an apostrophe with a plural form). Is this the sort of thing that we really want to enforce in this context?