That's the process's own name. OP seems to be trying to find the name of a child that he forked off.

Yes indeed, and the forked-off process - is a process, with its name residing in $0. The parent wants to know it for some reason. It could:

1. ask the kernel almighty
2. ask his offspring to tell, and listen to what it says.

Solving the first would be the "more general case" you mention. Solving the second possibly involves inter process communication (perlipc).