WordPress - Share a custom taxonomy with multiple post types in place of a single $object_type.

shared-taxonomies.php

add_action( 'init', 'create_my_taxonomies', 0 );
function create_my_taxonomies() {
    register_taxonomy(
        'type',
        array('page, 'post', 'posttype1','posttype2'), //An array of post types that share this taxonomy
        array(
            'labels' => array(
                'name' => 'Product Type',
                'add_new_item' => 'Add New Product Type',
                'new_item_name' => "New Product Type"
            ),
            'show_ui' => true,
            'show_tagcloud' => false,
            'hierarchical' => true
        )
    );
}