Na página bash
man:
-c If the -c option is present, then commands are read from the first non-option argument command_string. If there are argu‐ ments after the command_string, they are assigned to the positional parameters, starting with $0.
E mais abaixo:
If arguments remain after option processing, and neither the -c nor the -s option has been supplied, the first argument is assumed to be the name of a file containing shell commands. If bash is invoked in this fashion, $0 is set to the name of the file, and the positional parame‐ ters are set to the remaining arguments.
Em resumo, o comportamento de $0
varia dependendo de como o bash é invocado.