The man page for perlvar says:

   On some operating systems assigning to "$0" modifies
   the argument area that the ps program sees.
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The emphasis being on some

Printing $0 in your program means nothing. That's the value that your program sees. If, as a side effect, the value ps sees is also modified, you can bet that Perl isn't querying that ps value to show you what it has in $0.