get term slug from current posts’s custom taxonomy

This gets the terms:

$terms = get_the_terms( get_the_ID(), 'nieuwssoort');

Then you check the result:

 $terms_ids = [];//you can set a default id here
    if(false !== $terms && !is_wp_error($terms)){
        foreach($terms as $term){
             $terms_ids[] = $term->term_id;
         }
    }

We check the result just in case.

Now, we have the values, let get them into the arguments:

$args = array( 
          'post_type'       => 'nieuws',
          'orderby'         => 'date',
          'post__not_in'    => array($currentID),
          'tax_query' => array(
                           array(
                            'taxonomy' => 'nieuwssoort',            
                            'terms'    => $terms_ids
                           ),
          ),

          'posts_per_page'  => '7',
      );

I can’t test the code right now, but I think it should work.