How to add a shortcode function that returns the taxonomy slug of the actual post within the loop

get_the_terms() will always return an array; but because you have only one ‘project_category’, you can simply use its first element:

add_shortcode( 'return_taxonomy_slug', 'my_shortcode_return_taxonomy_slug' );
function my_shortcode_return_taxonomy_slug() {
  $terms = get_the_terms( get_the_ID(), 'project_category');
  return $terms[0]->slug;
}