I know of no way to create tagging groups. However...
Since you have only one post, I wonder if you are new to MC. I hope you realize that when you right-click in the tag pane, you can select which tags to show and which to hide, and that these choices persist (per media type), so that you only have to do it once. I also use many custom fields, and I've enabled the ones I want and hidden the ones I don't, and this has worked fine for me for years. If you're needing to actively edit more tags than can appear on your screen without scrolling, that is a very burdensome workflow.
Another option is to create a view, which shows as columns the tags you are interested in. Then, in the tag pane, you can right-click and select "Show tags in current view" and it will show you just those tags. You can of course also edit tags directly from the file list in the view. I have special "tagging views" setup for exactly this purpose.
Since you can create as many such views as you like, each with a different set of tags, this would in a way function as the tag groups you describe; you'd just have to enter that particular view to be able to select that particular set of tags.
Good luck...