/**
 * Set new user\'s display name as their login.
 * 
 * @link http://wordpress.stackexchange.com/q/138034/1685
 * 
 * @param int $user_id
 */
function wpse_138034_display_name_as_login( $user_id ) {
    if ( $user = get_user_by( \'id\', $user_id ) ) {
        // Prevent infinite loop.
        remove_action( \'user_register\', __function__ );

        wp_update_user(
            array(
                \'ID\' => $user_id,
                \'display_name\' => $user->user_login,
            )
        );

        // All done, restore.
        add_action( \'user_register\', __function__ );
    }
}

add_action( \'user_register\', \'wpse_138034_display_name_as_login\' );