Add term for custom taxonomy from front end

Your code doesn’t assign the terms to the post. You can do this with wp_set_post_terms() which does:

Set terms for a post.

Simplified usage example:

$pet_cat_ term = $_POST['pet_category'];  
wp_set_post_terms( $post_id, $pet_cat_ term, 'pet-category' );

Additional note: use the id as value for your term select(s), to prevent problems, i.e. because for hierarchical terms you have to do it anyways.