How to make a custom taxonomy selectable in post publish area?

The reason it does not show up automatically is because it requires REST enabled to display. By default, show_in_rest is set to false when registering taxonomies and post types. There’s currently a Trac Ticket #42785 open to change this default functionality.

So in short when registering a taxonomy while using the Block Editor you’ll need to explicitly say: 'show_in_rest' => true

function all_about() {
    register_taxonomy( 'all_about', 
        array('post'), /* if you change the name of register_post_type( 'custom_type', then you have to change this */
        array(
            'hierarchical' => false,    /* if this is false, it acts like tags */                
            'labels' => array(
                'name' => 'Categorization', /* name of the custom taxonomy */
                'singular_name' => 'Name', /* single taxonomy name */
                'search_items' => ' Search', /* search title for taxomony */
                'all_items' => 'All items', /* all title for taxonomies */
                'parent_item' => 'Parent item', /* parent title for taxonomy */
                'parent_item_colon' => 'Parent item colon:', /* parent taxonomy title */
                'edit_item' => 'Edit', /* edit custom taxonomy title */
                'update_item' => 'Update', /* update title for taxonomy */
                'add_new_item' => 'Add new', /* add new title for taxonomy */
                'new_item_name' => 'New title' /* name title for taxonomy */
            ),
            'show_ui' => true,
            'query_var' => true,
            'rewrite' => array( 'slug' => 'all-about', 'with_front' => false ),
            'show_in_nav_menus' => true,
            'show_tagcloud' => true,
            'show_in_quick_edit' => true,
            'show_in_rest' => true,
        )
    );
}
add_action( 'init', 'all_about' );