The 'Victorians' were fond of using umbilical cord clamps , and sterility was not a big issue with such clamps (used externally , post birth) .
If this is indeed such a clamp , I could see string/cord/yarn hooked/tied upon the ends to keep it in place (& firmly clamped) on the baby for awhile .
It is something of special purpose , it seems to me , so as to be stored in such a nice case (whatever the heck it turns out to be) .... also , the bellows design of the case may have been designed to prevent breakage (early-day 'shock absorber') of the tool housed within it .