Display latest x posts from all categories in Custom Post Type/Taxonomy

The category parameters does not work with custom taxonomies. You need to use a tax_query instead.

In your current code, replace (which is in any case also wrongly used, category_name takes the slug, not name)

'category_name' => $category->name

with

'post_type' => 'case-studies',
'tax_query' => array(
    array(
        'taxonomy' => $tax
        'terms' => $category->term_id,
        'include_children' => false
    )
),