That you want the file descriptor dup()ed, i.e. get a new file descriptor that refers to the same output stream as STDERR and a Perl-level file handle layered on top of that.
As the open() perldoc also explains, this is even better written with an '=' after the ampersand:
open my $save_out, '>&=', \*STDOUT or die "Can't fdopen STDOUT: $!";
open STDOUT, '>&=', $save_out or die "Can't restore STDOUT: $!";
This avoids creating an all new file descriptor but reuses the system's for a new Perl file handle.