Mah Jong and other game cases are typically constructed of dovetailed wood sides and cardboard. The joints are glued and the covering applied with hide glue. Hide glue softens with heat and moisture, so it is unusual to find paper-covered cases intact today. If covered in oilcloth, they tend to hold up a bit better.
To repair old cases, I use wood glue for the frame. But for the covering I use a mixture of wheat paste and PVA - very similar to that used by bookbinders.
For plastics, I have found Duco cement (nitrocellulose) to be best for celluloid (cellulose nitrate). The method used by guitar makers is to dissolve celluloid in acetone, and use that mixture as an adhesive. Too tricky for me. For other plastics, especially catalin, I use a slow-setting epoxy with good flexibility.