His body knows even if his mind tries to forget: the eye isn't his. It's a niggling remembrance of what he's lost, a scar that won't fade.
Everyone in the village carries things around with them, physical and mental, but Kakashi's is the most apparent. It's heavy, this reminder.
His students have scars too. He recalls finding Sakura, hair shorn off and green eyes rimmed with red and the constantly sad lines of Naruto's face.
The difference is that he knows that they will overcome with time. He's too old for that. He'll carry it with him until he dies.