Show related posts based of current ACF field name in a single page post (a loop within loop)

You don’t have a meta_value in your query.

From your code I can’t tell what you’re doing with the meta values but you should be able to do this:

// Store this BEFORE you go into the second loop.
$category = get_field('category');

// The arguments for the second loop
'post_type'   => 'case-studies',
'showposts'  => 3,
'meta_key'    => 'category',
'meta_value'   => $category,
'orderby'     => 'rand'

So what you’re doing there is matching the value of the category meta key.

Ideally you would be using a taxonomy to do the above – that’s what they are there for.

http://codex.wordpress.org/Function_Reference/register_taxonomy