Защо кодът в успешното извикване на AJAX не работи?

success е името на callback функция, а не promise .

Трябва да използвате:

$.ajax({
    type: "POST",
    url: "<?php echo site_url('con_product/ins_product'); ?>",
    data: data,
    success: function() {
        alert("Products list is ready to be printed");
        window.open("<?php echo site_url('con_product/print_product'); ?>","_blank");
        window.open("<?php echo site_url('con_product/form_product'); ?>","_self");
    }
});

Или done обещание:

$.ajax({
    type: "POST",
    url: "<?php echo site_url('con_product/ins_product'); ?>",
    data: data
}).done(function() {
    alert("Products list is ready to be printed");
    window.open("<?php echo site_url('con_product/print_product'); ?>","_blank");
    window.open("<?php echo site_url('con_product/form_product'); ?>","_self");
});