How to get dropdown instance value in WordPress custom Widget

The cat_drop setting is empty because your drop-down menu, or the select field doesn’t have the proper name value, so PHP isn’t receiving the selected option/value. So to fix the issue, just add the proper name value to the select field:

<!-- Wrapped for brevity. -->
<select
  name="<?php echo $this->get_field_name( 'cat_drop' ); ?>"
  id="<?php echo $this->get_field_id( 'cat_drop' ); ?>"
>

But then, there are other issues in your code:

1. In the cat_drop drop-down menu, the options should be the term slug, but you currently set it to the term name. So be sure to use the term slug instead:

   <!-- Note the $item->slug -->
   <option value="<?php echo $item->slug;?>"><?php echo esc_html( $item->name ); ?></option>
   

2. In the widget display callback (widget()), you’re overriding the $args variable when you do $args = array('post_type' => 'game', 'gamecategory' =>$selected_cat); and that results in a PHP notice with the echo $args['after_widget']; at the end of the function, because the $args['after_widget'] is now undefined. And when it’s undefined, the widget container will not be closed and that could result in layout issues, despite browsers are basically “smart” and would-auto close the container.

   So you should rename one of the $args, maybe the one that you use with WP_Query.

3. Using esc_html__() on an empty string is pointless: esc_html__( '', 'prisma' ). So change those to just ''.

4. This is just a personal note.. Could you please, improve the formatting of your code? E.g. Get rid of those unnecessary blank lines and use tab for indentation. Because a good code is not just one that works, but also clean because it would be easy for other developers to maintain or work with the code. 🙂