My question back to you is: why did you create the tag groups?
Have a look at this /tags page. It’s showing the tags according to their relationship to the categories. The only use for tag groups right now is to restrict their use in categories, so we show the category-to-tags relationships on the /tags page. But it sounds like you haven’t defined those relationships.
I guess you created tag groups for cosmetic reasons, but we don’t currently have a way to show that.
It’s not possible either. How do you imagine differentiating the groups in the dropdown?