My Airbnb host in India told me prior to my arrival:
According to Indian law, we would require a copy of your passport.
Are Airbnb hosts required to obtain copies of the guests' passports in India?
Yes, this is correct - see Section 4 of https://www.airbnb.co.uk/help/article/1641/guidance-on-foreigner-registration--frro--process-for-india-hosts (mirror).
Any person hosting a foreign guest for monetary consideration in India, even for a few hours, should submit C-Forms.
The submission of C-Forms and maintenance of guest registers stems from the Foreigners Act 1946 and Registration of Foreigners Rules, 1992. These requirements extend across India, and apply to all persons who accommodate foreign guests for monetary consideration, including hotels, guest houses, lodging houses etc.
The C-Form mechanism helps the authorities locate and track foreigners in India to enhance security and safety. Failure to comply with reporting requirements could result in fines and imprisonment of up to 5 years. Section 4 states that a copy of the visitor’s visa and passport has to accompany C-Form. Additionally, in some localities, per the local office guidelines, the host may also be required to submit the host’s address proof again, such as a utility bill.