Yes. It’s the law. Art. 109 of the TULPS (Testo Unico Legge Pubblica Sicurezza=Uniform Public Security Law) which requires operators of hotels, vacation homes, etc. to communicate guests’ info (copies of passport or similar ID) to the police authorities within 24 hr from their arrival.
If your host doesn’t comply with the law he/she will incur a fine of 206€ and a prison term of up to 3 months in jail, therefore your cooperation will be deeply appreciated by the owner of your AirBnB.
(This art. 109 was modified many times over the years, most recently this past year, but actually it originally dates back to long before the terrorism years of the 1980s. It was first introduced by a Royal decree in June 1931 by the Mussolini administration).