No, it shouldn’t, you just need to do a little more work to get the term id
<?php
$taxonomy = 'product_cat';
$queried_object = get_queried_object();
$post_id = $queried_object->ID;
$terms = wp_get_post_terms($post_id, $taxonomy, array('fields' => 'ids'));
$term_id = $terms[0];