Called inrō, the case is in fact a set of several compartments, held closed by a silk cord and worn tied to the obi. The inrō is still worn today on special occasions, as you can see in this photo, where Awa Odori dancers are carrying small but beautiful inrō cases...
EXIF info:
Nikon D90
Lens: VR 18-55mm F/3.5-5.6G
Focal Length: 31mm
Aperture: F/4
Shutter Speed: 1/60s
ISO Sensitivity: ISO 500