Nintendo 3DS Music always had the option, as did DSi Music before it - plugging in headphones was the trigger. Kirby: Triple Deluxe also allows you to listen in Sleep Mode, on by default and untoggleable - I imagine Sakurai put the toggle in Smash to ensure battery can be conserved whenever players stop listening to the music and put their 3DS in their bag or something like that, because the system doesn't actually go into a "proper" Sleep Mode when music plays through - the screen simply turns off while the software continues operation, and that goes for KTDX as well.
NNID: TouyaShiro_lucia [Currently playing - The Legend of Zelda: The Wind Waker HD. "Where are you, Mapfish!?"]