Vernier caliper jaws shape purpose
The jaws have a knife edge for measuring the distance between two holes, which is needed when measuring the distance between hole centres.

If the jaws didn't have a knife-edge tip, the width of the tip would hold the caliper away from the hole at two points and the measurement would be wrong.




A - knife edge on inside jaws for measuring inside holes
B - recess to give clearance and define measuring surface
C - ditto
D - rectangular edge for measuring flat (or convex) surfaces. This edge is usually lapped flat, an expensive precision operation that cannot be easily performed up against an internal corner, hence the recessed corner.
E - knife edge for measuring in holes (or similar surfaces that are concave in one dimension) or where access is limited.




The recesses for clearance are found on other tools where you need to prevent sharp corners (or burrs) on the object from contacting a radiussed inner corner on the measuring tool.

Note the recess in the corner of this engineer's square





There are a wide range of jaw shapes available for specialised purposes. 


Source - Mitutoyo

The most common jaw shape is probably a compromise that is useful in a wide range of situations.