my $filename = $ARGV[0];
my $nb_lines = 0;
open (my $file, $filename) or die "Can't open '$filename': $!";
my $hdl_sauv;
while (<$file>) {
    $nb_lines++ ;
    if ($nb_lines == 5) {
    open ($hdl_sauv, "<",$file) || die "cannot open $file\n";
    }
}

my $saved_line = <$hdl_sauv>;
print "$saved_line\n";
[download]

It looks like you want something like this:

my $filename = $ARGV[ 0 ] or die "Usage: $0 file\n";
open my $file, $filename or die "Can't open '$filename': $!";
while ( <$file> ) {
    last if $. == 5;
}

print;
[download]

I am trying to make a copy of the filehandle, that is *not* sharing file position.

Your code does not make a copy of the filehandle, nor does it share the file position.    It tries to open the previous filehandle as if it were a file name.    If you had opened $filename instead of $file you would have a different filehandle for the same file, but that still wold not solve your problem as it would be pointed at the beginning of the file.

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