Ok, so I’ve figured it out. Turns out, the plugin I used to clone an existing site on a multisite configuration, did not copy wp_woocommerce_termmeta table. Just copy its structure as wp_2_woocommerce_termmeta (replace 2 with your actual site id) and it will work as expected.
They do save. They don’t get displayed unless I sort by name or description or anything other than order. Because there is no order meta value created for them. I can create them, but I can’t use them, because everywhere else they are sorted by order, so nothing is displayed.