I had exactly the same with an inexperienced user who joined airbnb just to book my apartment.
That's what I thought too, they gave it to Airbnb thinking I could access it.
Yes Cynthia and Chris you can ask to see it upon arrival.
In my area we have to rewrite all the information and send it to the police.
In my experience there is so much to discuss at checkin, people are tired after their travels and eager to get on with their holiday, and this becomes a huge burden. People don't want to sit for 3 minutes and rewrite their passport data on a piece of paper.
We ask for it ahead of time to streamline the process and also for our admin, the person who welcomes the guests is not the person who does the police registration task.
So there is a valid reason for some hosts to ask for it ahead of time.
Yet another reason Airbnb should allow the sending of attachments between hosts and guests once the booking is made.