The other party would have to proof that you're liable, in other words that you personally have shared the file and not another person who may have stayed with you in the Airbnb. I would reckon that's nearly impossible.

If I was to make a payment to their PayPal account out of the goodness of my heart/to assuage my sense of guilt, would it constitute an admission of liability?

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The AirBnB host could also relatively easy deny that they have shared the file. They don't have to name the responsible person - they would only have to show that it is plausible that another person has commited the infringement.

I wondered about this, but they seemed to think they had no option but to pay. Apparently under German law the owner of the internet connection is responsible for securing it, and ensuring that it's not misused by anyone that has the password. I haven't worked out yet how it changes the moral landscape if they could easily have avoided the fine