I don't get it, why do you include | in the delimiter? Do you also want to count that? it has no special meaning in tr.

I'd write that as

sub count_separators {

    my $value = shift;
    my $delim = shift;
    my %map = (
         '('   => '()',
         '{'   => '{}',
         '"'   => '"';
    );
    my $tr = $map{$delim} or die "Not a valid separator";
   
    return eval "\$value =~ tr/$tr/;"
}
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(untested).