How to list custom tags too, to wp-list-table table?

You just need to add 'show_admin_column' => true to your taxonomy parameters:

‘show_admin_column’
(bool) Whether to display a column for the taxonomy on its post type listing screens. Default false.

So,

register_taxonomy(
    "custom_CUSTOM_tag", //taxonomy
    "post",
    array(
        'hierarchical'      => false,
        'label'             => 'title',
        'singular_name'     => 'title',
        'rewrite'           => true,
        'query_var'         => true,
        'labels'            => [
            'add_new_item' => 'new',
        ],
        'public'            => true,
        // Add this line:
        'show_admin_column' => true,
    )
);

Update

so far good, but it adds another column. Cant I append tags into the
“built-in” column?

Yes, you can, and one way is by using the post_column_taxonomy_links hook. Here’s an example which appends the custom tags (or terms in the title taxonomy) to the default “Tags” column in the posts list table:

But first off, please remove the 'show_admin_column' => true, from your taxonomy parameters.

add_filter( 'post_column_taxonomy_links', 'my_post_column_taxonomy_links', 10, 2 );
function my_post_column_taxonomy_links( $term_links, $taxonomy ) {
    global $post;

    if ( 'post_tag' === $taxonomy && is_object( $post ) ) {
        $terms = get_the_terms( $post->ID, 'title' ); // be sure the taxonomy is correct

        if ( is_array( $terms ) ) {
            foreach ( $terms as $term ) {
                $term_links[] = sprintf( '<a href="%s">%s</a>',
                    esc_url( get_edit_term_link( $term->term_id, $term->taxonomy, 'post' ) ),
                    esc_html( $term->name )
                );
            }
        }
    }

    return $term_links;
}

And BTW, that singular_name actually belongs in the labels item, so you should correct it. 🙂