If you look at the source code of get_the_category() you see that this is not much more than a call to get_the_terms ( $id, 'category' ), where $id is the current post id.
So, what you need is get_the_terms ( get_the_ID(), 'game' ). You need get_the_ID, because the ID is not passed to the shortcode as it is to get_the_category.