WordPress - Create New Custom Taxonomy

// create a new taxonomy we're calling 'layout'
// if you call the new taxonomy 'format' and you have support
// for post formats enabled the taxonomy will not be visible or work
function custom_post_formats_taxonomies() {
    // Add new taxonomy, make it hierarchical (like categories)
    $labels = array(
        'name'              => _x( 'Layouts', 'taxonomy general name', 'textdomain' ),
        'singular_name'     => _x( 'Layout', 'taxonomy singular name', 'textdomain' ),
        'search_items'      => __( 'Search Layouts', 'textdomain' ),
        'all_items'         => __( 'All Layouts', 'textdomain' ),
        'parent_item'       => __( 'Parent Layout', 'textdomain' ),
        'parent_item_colon' => __( 'Parent Layout:', 'textdomain' ),
        'edit_item'         => __( 'Edit Layout', 'textdomain' ),
        'update_item'       => __( 'Update Layout', 'textdomain' ),
        'add_new_item'      => __( 'Add New Layout', 'textdomain' ),
        'new_item_name'     => __( 'New Layout Name', 'textdomain' ),
        'menu_name'         => __( 'Format', 'textdomain' ),
    );

    $args = array(
        'hierarchical'      => true,
        'labels'            => $labels,
        'show_ui'           => true,
        'show_admin_column' => true,
        'query_var'         => true,
        'rewrite'           => array( 'slug' => 'layout' ),
        'capabilities' => array(
            'manage_terms' => '',
            'edit_terms' => '',
            'delete_terms' => '',
            'assign_terms' => 'edit_posts'
        ),
        'public' => true,
        'show_in_nav_menus' => true,
        'show_tagcloud' => true,
    );
    register_taxonomy( 'layout', array( 'post' ), $args ); // our new 'format' taxonomy
}

// hook into the init action and call custom_post_formats_taxonomies when it fires
add_action( 'init', 'custom_post_formats_taxonomies', 0 );

// programmatically create a few format terms
function example_insert_default_layout() { // later we'll define this as our default, so all posts have to have at least one format
    wp_insert_term(
        'Default',
        'layout',
        array(
          'description' => '',
          'slug'        => 'default'
        )
    );
}
add_action( 'init', 'example_insert_default_layout' );

// repeat the following 11 lines for each format you want
function example_insert_map_layout() {
    wp_insert_term(
        'Map', // change this to
        'layout',
        array(
          'description' => 'Adds a large map to the top of your post.',
          'slug'        => 'map'
        )
    );
}
add_action( 'init', 'example_insert_map_layout' );

// make sure there's a default Format type and that it's chosen if they didn't choose one
function moseyhome_default_layout_term( $post_id, $post ) {
    if ( 'publish' === $post->post_status ) {
        $defaults = array(
            'layout' => 'default' // change 'default' to whatever term slug you created above that you want to be the default
            );
        $taxonomies = get_object_taxonomies( $post->post_type );
        foreach ( (array) $taxonomies as $taxonomy ) {
            $terms = wp_get_post_terms( $post_id, $taxonomy );
            if ( empty( $terms ) && array_key_exists( $taxonomy, $defaults ) ) {
                wp_set_object_terms( $post_id, $defaults[$taxonomy], $taxonomy );
            }
        }
    }
}
add_action( 'save_post', 'moseyhome_default_layout_term', 100, 2 );

// replace checkboxes for the format taxonomy with radio buttons and a custom meta box
function wpse_139269_term_radio_checklist( $args ) {
    if ( ! empty( $args['taxonomy'] ) && $args['taxonomy'] === 'layout' ) {
        if ( empty( $args['walker'] ) || is_a( $args['walker'], 'Walker' ) ) { // Don't override 3rd party walkers.
            if ( ! class_exists( 'WPSE_139269_Walker_Category_Radio_Checklist' ) ) {
                class WPSE_139269_Walker_Category_Radio_Checklist extends Walker_Category_Checklist {
                    function walk( $elements, $max_depth, $args = array() ) {
                        $output = parent::walk( $elements, $max_depth, $args );
                        $output = str_replace(
                            array( 'type="checkbox"', "type='checkbox'" ),
                            array( 'type="radio"', "type='radio'" ),
                            $output
                        );
                        return $output;
                    }
                }
            }
            $args['walker'] = new WPSE_139269_Walker_Category_Radio_Checklist;
        }
    }
    return $args;
}

add_filter( 'wp_terms_checklist_args', 'wpse_139269_term_radio_checklist' );