Use atributes of shortcode in get template part

The $atts in your template part (templates/show-taxonomy.php) that you passed to shortcode_atts() is not defined, so you need to define it by passing it via the third parameter for get_template_part() like this:

get_template_part( 'templates/show-taxonomy', '', array( 'atts' => $atts ) );

Then in the template part, use $args['atts'] to access the shortcode parameters:

$atts = shortcode_atts( array(
    'custom_taxonomy' => '',
), $args['atts'], 'atributes' ); // here, use $args['atts'] and not $atts

And actually, in the above shortcode_atts() call, the third parameter should be cat_widgets (i.e. the 1st parameter for add_shortcode()) and not atributes. 🙂