Leanne, It sounds like what you have is something known (though
not by the general public) as a "mosque." Mosques are multi-doored,
tower-like music boxes whose compartments are designed to hold small
objects, such as perfumes, liqueur bottles, cigarettes, jewelry, etc.
I believe Reuge, the one surviving Swiss music box manufacturer,
still makes mosques. You will see some examples of the type at