I did have almost the exact code for menu as you have written, however as I said, it doesn't work, even when exactly the yours. Try for yourself... but never mind. I solved the problem with very simple jQuery function. With the same function I also solved highlighting the blog menu item when some post is shown, and paging of blog which is also home page. So, Blog gets highlighted when on home page, when on paged blog/home page, and on post page. I'll put my web-site on GitHub so that other folks can help themselves if encountering similar problem.
Regarding tag keys/names, I did not find any appropriate discussion. This solution, if it was even aimed at similar problem, simply doesn't work. In this discussion, usage of tag keys was shown, not names. I haven't found any other recent discussion, for current version of Hugo.
If I'm missing something, please let me know.