@Annette33 That's not entirely true. Schengen has nothing to do with it.
I'm from Denmark and I went to Milan, Italy the other week and the host also asked for my passport or drivers license for him to provide to the local authorities.
In the this situation @Ahmed25 you can just fill out the form without hesitating. It's legit and the host simply follows the local laws regarding rentals.
It merely depends on the country's law and regulations. We don't ask for such infomation in Denmark but some countries do.
Schengen has to do with EU citizents being able to move across borders without being treated differently than the country's citizents. If you are an EU citizen you can cross borders within the EU without restrictions and travel completely freely without a passport. But you will need ID so many of us travel with passports anyway - and airlines usually always require a passport but not all.
Although many countries have temporarily suspended the agreement due to the refugee crisis but it has nothing to do with rental houses nor Airbnb listings inside or outside EU.