The spindle is what the switch is mounted on. If the switch is slightly loose it won't turn the spindle and the camera will stay in the multi-exposure setting.
In the two attachments below you can see the multi-exposure switch on my camera. In the first it has the rubberised centre cap in place, in the second the centre cap is removed revealing the end of the spindle. If you turn the switch and the spindle does not move then you have a loose switch.
The centre cap is kept in place with a small dab of what looks like PVA glue.