If I duplicate a Woocommerce product, the custom taxomomy markup is empty.

Impossible for me to say, as we don’t offer any sort of post duplication with our plugin.

If this functionality is something specific for WooCommerce based product posts, I wonder if that only duplicates content that is commonly known to WooCommerce, like Woo’s own registered taxonomies, and doesn’t know to work with any custom added taxonomies.

If it’s just a generic post duplicator plugin like https://www.ads-software.com/plugins/duplicate-post/ for example, I would imagine they’d copy EVERYTHING, but not guaranteed as I haven’t tinkered with that one that much, especially with WooCommerce.