I agree it is a caliper type poising tool. These are pretty common and I
probably have at least a dozen different types.

For a watch to run accurately the balance wheel need to be in poise, all
points of the rim must be of equal mass or weight.  The balance wheel
pivots go into small depressions in the faces of the caliper tips and
the wheel can then be spun to see if there are heavy spots and to make
sure the rim of the wheel spins true.

Troy