Enable Shortcode for Custom Wordress Widgets

Your shortcode function will register widget, will not display on a page as a shortcode.

To show widget on a page/post you need to use the_widget( 'test_widget', $instance, $args ); function.

Codex documentation is here

For example, in your code

// Display the widget
function test_show_widget() {
    the_widget( 'test_widget' );
}

add_shortcode('test_widget', 'test_show_widget');