Custom shortcode is not working in text widget

Shortcodes need to return data, not echo it – use output buffering to capture the output of the include and return it:

function related_category_sidebar() {
    ob_start();
    include WP_PLUGIN_DIR . '/sabai-directory/assets/templates/template_related_category.php';
    return ob_get_clean();
}

Then you will also need to do as Charles suggested and enable shortcodes for text widgets:

add_filter( 'widget_text', 'do_shortcode', 11 );