我已经找到了答案，我完全误解了标准帖子类型和自定义帖子类型之间的区别，以及它们的分类法。我发现了几篇非常有用的文章，下面的代码就是我最终得到的，还有一些我觉得有用的文章。我希望其他人觉得这很有用。

// Load Custom Post Type 
function add_directory() {
    $labels = array( 
    \'name\'               => __( \'Businesses\', \'text_domain\' ),
        \'singular_name\'      => __( \'Business\', \'text_domain\' ),
        \'add_new\'            => __( \'Add New Business\', \'${4:Name}\', \'text_domain\' ),
        \'add_new_item\'       => __( \'Add New Business\', \'text_domain}\' ),
        \'edit_item\'          => __( \'Edit Business\', \'text_domain\' ),
        \'new_item\'           => __( \'New Business\', \'text_domain\' ),
        \'view_item\'          => __( \'View Business\', \'text_domain\' ),
        \'search_items\'       => __( \'Search Businesses\', \'text_domain\' ),
        \'not_found\'          => __( \'No Businesses found\', \'text_domain\' ),
        \'not_found_in_trash\' => __( \'No Businesses found in Trash\', \'text_domain\' ),
        \'parent_item_colon\'  => __( \'Parent Business:\', \'text_domain\' ),
        \'menu_name\'          => __( \'Business Directory\', \'text_domain\' ),
    );
    $args = array( 
        \'labels\'              => $labels,
        \'hierarchical\'        => true,
        \'description\'         => \'description\',
        \'taxonomies\'          => array( \'biz-cat\' ),
        \'public\'              => true,
        \'show_ui\'             => true,
        \'show_in_menu\'        => true,
        \'menu_position\'       => 5,
        //\'menu_icon\'         => \'\',
        \'show_in_nav_menus\'   => true,
        \'publicly_queryable\'  => true,
        \'exclude_from_search\' => false,
        \'has_archive\'         => true,
        \'query_var\'           => true,
        \'can_export\'          => true,
        \'rewrite\'             => true,
        \'capability_type\'     => \'post\', 
        \'supports\'            => array( \'title\', \'editor\', \'author\', \'thumbnail\', \'custom-fields\', \'revisions\', \'post-formats\' ),
    );
    register_post_type( \'bizdirectory\', $args );
}
add_action( \'init\', \'add_directory\' );

// Create Custom Taxonomy
function directory_create_taxonomies() 
{
    register_taxonomy( \'biz-cat\', array( \'bizdirectory\' ), array(
        \'hierarchical\' => true,
        \'label\' => \'Business Categories\',
        \'singular_name\' => \'Business Category\',
        \'show_ui\' => true,
        \'query_var\' => true,
        \'rewrite\' => array( \'slug\' => \'biz-cat\' )
    ));
}
add_action( \'init\', \'directory_create_taxonomies\', 0 );

// \'bizdirectory\' is the registered post type name
function directory_columns($defaults) {
    // \'biz-cat\' is the registered taxonomy name
    $defaults[\'biz-cat\'] = \'Business Category\';
    return $defaults;
}
function directory_custom_column($column_name, $post_id) {
    $taxonomy = $column_name;
    $post_type = get_post_type($post_id);
    $terms = get_the_terms($post_id, $taxonomy);

    if ( !empty($terms) ) {
        foreach ( $terms as $term )
            $post_terms[] = "<a href=\'edit.php?post_type={$post_type}&{$taxonomy}={$term->slug}\'> " . esc_html(sanitize_term_field(\'name\', $term->name, $term->term_id, $taxonomy, \'edit\')) . "</a>";
        echo join( \', \', $post_terms );
    }
    else echo \'<i>Not assigned.</i>\';
}
add_filter( \'manage_project_posts_columns\', \'directory_columns\' );
add_action( \'manage_project_posts_custom_column\', \'directory_custom_column\', 10, 2 );