I think in this case it's a personal decision on your part. The guest probably isn't expecting a refund (if not part of your cancellation policy) but may ask. So you don't have to of course.
But then there's the karma answer. I personally would probably offer to refund the second night. I always appreciate an unexpected night to myself even if it's not paid. And it would seem fair to me. That's just how I look at my Airbnb business - I don't want to get too caught up about the money then things like this become a negative. I prefer to offer than "give in" to a request/demand.
Hope it works out well...