The narrowing allows you to avoid the rounding of an edge (e.g. of a lip) or the rounding of a conical hole bottom (e.g. of a drilled hole).
Or you can opt to deliberately measure the depth at the edge or bottom before any rounding.
This is also why the narrowed protrusion is offset to one side, e.g. in your picture it is aligned with the bottom of the gauge, and not centred.
A drilled hole's depth is generally the depth at full diameter, so it's measured along the wall. The blind hole is measured at its flat bottom, and any rounding (perhaps due to a fillet of grime or crud) at the bottom corner should be avoided. Important thing is to be aware of these subtleties in order to make the measurement that matters for the case at hand.
Depending on how it's held, you can aim for an offset or not and either way run it tight along the edge for a straight measurement.


Ref: https://www.stefanelli.eng.br/en/use-vernier-caliper/