First, what the United States Supreme Court said. What the United States Supreme Court held in Hiibel v. Sixth Judicial Dist. Court (2004) 542 U.S. 177, was that a state could make it a crime for a person to refuse to identify himself (i.e., tell the officer his name and address) when lawfully detained for criminal activity. Note that the Supreme Court did NOT say that any kind of identification papers could be required, nor did they say that police officers could ordinarily arrest someone for refusing to identify himself absent a state law permitting that arrest. There is no law in the United States requiring everybody to carry ID, at least not yet.
There is NO law in California requiring anybody to carry identification. There is no law making it illegal for anyone (even someone lawfully detained) to fail to have identification papers or to refuse to identify himself (there was such a law, which was declared unconstitutional). Thus, Hiibel is of no effect in California, since there is no comparable law there. (It is, however, a crime to give a FALSE identification.)
A person CANNOT be arrested just for failing to identify himself or failing to have ID, even with a lawful detention. It is NOT interfering with an officer. The only effect of not having ID occurs if a police officer has probable cause to believe an arrestee has committed a criminal offense. A police officer who could otherwise give an arrestee a citation to appear would instead take the person into custody to appear before a magistrate. But this is ONLY if the officer has probable cause to believe the person has committed a crime--NOT just because the person did not have ID.
Of course, one must have identification in his or her possession when driving, and a police officer obviously can demand to see a drivers license from any driver lawfully detained.
30+ years as a criminal defense attorney