Occasionally I'll run into users, usually new ones, who don't understand what certain tags mean (for example, marking a high-school-level Euclidean geometry question with 'differential-geometry'). Usually this isn't a problem; I change the tags, and that's the end of it. At the moment, I'm dealing with a question where the user insists on adding a completely inapplicable tag to the question and repeatedly reverts my edits. I've added a comment to that effect, but it was dismissed, and the edits continue. If the user insists on using an incorrect tag, is there anything further I can do, or this is an instance where I just have to walk away?
Don't engage in edit wars.
If your help with tagging is not appreciated, it's easy to find another place where it will be.
And this example is not so clear cut. A circle is a smooth manifold. Rouché's theorem is an argument by smooth homotopy. The author mentions that more general degree theory facts may be used, and so tagged with differential-geometry and differential-topology. Yes, it's a stretch; and yes, a retag was appropriate. But a forceful retag after the author's rollback was not.
As an aside, your comment
differential geometry (the study of Riemannian metrics on manifolds and associated invariants)
isn't a correct definition of differential geometry either.