I had a Link Category that happened to be exactly the same word as a Tag. Let's say is was "beer" just as an example. I wanted to edit either the Link Category or the Tag in order to make them different. Whenever I'd go into the Tag manger admin page and edit the tag, the Link Category would also get changed, and vice-versa! I.E. if I edited the "beer" Tag to read "beers", then would I looked at my Link Categories, the tag there would also have been changed to read "beers". If I edited the Link Category to change it back to "beer", then the Tag would also automatically get changed back to "beer"!
I noticed that the tag_id and cat_id were the same, which I thought might have been a source of some of the confusion to WP. So, I created a new Link Category called "Beers", and then re-assigned my Links from the "beer" Link Category to the "Beers" Link Category, and then deleted the Link Category "beer". This worked fine.
However, now if I try to rename the Link Category from "Beers" to "Beer", after I click the Update button, it just says "Category not updated." It appears that I can't have a Link Category with the same name as a Tag. Editing a different Link Category works fine. Why should WP care if my Tag and Link Category names overlap?