There is a filter called register_taxonomy_args in WP v4.4 that’s a big help here. This tested and working example assumes that the post types schools and productions have already been created.

/**
 * Filter the arguments for registering a taxonomy.
 *
 * @since 4.4.0
 *
 * @param array  $args        Array of arguments for registering a taxonomy.
 * @param string $taxonomy    Taxonomy key.
 * @param array  $object_type Array of names of object types for the taxonomy.
 */
add_filter( 'register_taxonomy_args', 'wpse238461_taxonomy_school_subjects_title_changer', 10, 3 );
function wpse238461_taxonomy_school_subjects_title_changer( $args, $taxonomy, $object_type ) {

    // We'll need to figure out what kind of post type we're looking at, and it's too early for other methods.
    if ( ! isset( $_GET['post_type'] ) || ! $_GET['post_type'] ) {
        return $args;
    }
    $post_type = $_GET['post_type'];

    /*
    // Bonus code for anyone trying to do this on regular post or page edit screens,
    // $_GET['post_type'] is not set, so do something like this instead of/in addition to the above check

    // We'll need to figure out what kind of post type we're looking at, and it's too early for other methods.
    if ( ! isset( $_GET['post'] ) || ! $_GET['post'] ) {
        return $args;
    }

    // Get the post type (post or page), bail if we can't. 
    $post_type = get_post_type( $_GET['post'] );
    if ( ! $post_type ) {
        return $args;
    }   
    */

    // Make sure we're looking at the right taxonomy since this filter runs for all of them.
    if ( 'school_subjects' !== $taxonomy ) {
        return $args;
    }

    // Check if we're viewing the appropriate post type, and modify the label.
    if ( 'schools' === $post_type ) {
        $args['labels']['name']              = _x( 'Subject Wishlists', 'taxonomy general name', 'textdomain' );

        // Alternate example for when taxonomy is using the label argument instead of the entire labels argument:
        // $args['label'] = 'Subject Wishlist';
    }

    return $args;
}

// Taxonomy registration code
add_action( 'init', 'create_taxonomy_school_subjects' );
function create_taxonomy_school_subjects() {
    register_taxonomy(
        'school_subjects',
        array( 'schools', 'productions' ),
        array(
            //'label' => __( 'School Subjects' ),
            'labels' => array(
                'name'              => _x( 'School Subjects', 'taxonomy general name', 'textdomain' ),
                'singular_name'     => _x( 'School Subject', 'taxonomy singular name', 'textdomain' ),
                'search_items'      => __( 'Search School Subjects', 'textdomain' ),
                'all_items'         => __( 'All School Subjects', 'textdomain' ),
                'parent_item'       => __( 'Parent School Subject', 'textdomain' ),
                'parent_item_colon' => __( 'Parent School Subject:', 'textdomain' ),
                'edit_item'         => __( 'Edit School Subject', 'textdomain' ),
                'update_item'       => __( 'Update School Subject', 'textdomain' ),
                'add_new_item'      => __( 'Add New School Subject', 'textdomain' ),
                'new_item_name'     => __( 'New School Subject Name', 'textdomain' ),
                'menu_name'         => __( 'School Subjects', 'textdomain' ),
            ),
            'rewrite' => array( 'slug' => 'school_subjects' ),
            'hierarchical' => true,
        )
    );
}

School – Subject Wishlists

Production – School Subjects