Posts usually show on their single post view all tags that belong to that post. Since they are displayed by your theme, these tags are not aware of tag groups.
If you want to display your tags sorted into groups, you can use the “Post Tags” feature. When activated, we insert a table with groups and tags at the end of the post content. We use of course only tags belonging to that particular post. It is also possible to hide the default tags in most themes so that they won’t appear twice.
Since your theme is hard-coded, we cannot insert the tags at any position. It is only possible to change the order compared to content inserted by other plugins.
You can turn this feature on and configure it in the settings.