Can I register my address (Anmeldung) while staying in an Airbnb?

Only if the host cooperates. The registration office needs a landlord confirmation (Wohnungsgeberbestätigung, § 19 BMG) signed by the person providing the accommodation. Without it, the Bürgeramt will not register you at that address. Many Airbnb hosts refuse because registering guests can conflict with their own lease or with local short-term-rental rules; some hosts who rent legally do issue it, so ask before you book.

When you must register at all (Bundesmeldegesetz):

• Anyone who moves into a home in Germany has to register within two weeks of moving in (§ 17 BMG).
• If you come from abroad and are not yet registered anywhere in Germany, a temporary stay creates the duty to register only once it exceeds three months (§ 27 (2) BMG). For hotels and similar commercial accommodation the ordinary registration duty starts after six months, but a person without any German registration must register within two weeks once the stay exceeds three months (§ 29 BMG).

So a few weeks in an Airbnb while you look for a room does not by itself oblige you to register. The practical problem is that you usually need the registration earlier: for the residence permit, a bank account, the tax ID and enrolment at some universities. Plan for it:

• prefer temporary rooms or student halls whose provider confirms the Wohnungsgeberbestätigung in advance;
• as soon as you sign a proper lease or sublease, register within two weeks;
• do not use a friend's address without actually living there – incorrect registration is an administrative offence.

Source: §§ 17, 19, 27, 29 Bundesmeldegesetz (gesetze-im-internet.de).