People actually do this?

I usually give some indication, usually a shirt or some similar clear indication.

Generally I find that the way to accuse someone of being a furry is to wave a censer and point at them while decrying loudly "The Malleus Maleficarum hath told me sure signs of which thou art, foul furry! In the name of blessed St. Michael and St. Julian the Hospitaler I abjure thee!" If they tell you they're being "fursecuted" it's generally a sure sign. A "yew sure do talk mighty fancy" usually suggests you're barking up the wrong tree.

Yes, I think that is how I will handle this situation.