But can people in Japan tell you're Korean just by looking at your face?

I walked into a yakiniku restaurant,
and after they said "Irasshaimase" and showed me to my seat, they didn't even say a single word to me,
but they suddenly handed me a Korean menu?
Can they tell if someone is Korean or Japanese just by the sound of them breathing?
I mean, if I actually speak, I'm sure they'd be able to tell the difference.