As usual, JPM and silversvs have done a great job at addressing the legal issues contained in the original post as well as addressing some of the rampant misinformation that has cropped up here from well intended but misinformed citizens.
What this issue really boils down to is "satisfactory identification." This is not something that is defined in the code, but is a concept that (as with may legal interpretations) is in the eye of the beholder.
If you are LEGALLY DETAINED or ARRESTED by a peace officer (this includes a traffic stop or a stop for any code, ordinance, regulation or law), you can be compelled to provide SATISFACTORY IDENTIFICATION. This is generally held to be the information needed to complete a traffic citation that is supported by some legitimate documentation or other information that the officer feels adequately corroborates the information given.
Of course, the easy solution is to show a government issued identification card or license. That makes the process quick and painless and that is why I strongly suggest people carry their ID or DL when walking, riding a bicycle or going out and about in public. If you are stopped for ANY legal reason, you can be compelled to satisfactorily identify yourself.
In the past, when a government ID was not available, I have used a random combination of available documentation; paystubs, credit cards, school ID, vehicle registration and completed job applications, posted and unposted mail in addition to verbal assurances from uninvolved parties and descriptors from DMV records (height, weight, eye color, age) as well as other identifying information (scars, marks and tattoos) contained in court, booking and probation records or warrants. Many times I also ask questions regarding past addresses, past citations, past arrests, to confirm identity. Whatever it takes to satisfy or assure me that the person IS who they say they are.
Yes, I have arrested many, many people, bicyclists, pedestrians, skateboarders and go-peders, that have not had ID in their possession and either refused to give satisfactory information, were unable to give satisfactory information or gave false information. I have been doing this job for over 20 years and I can tell you, with computer systems and communications available today, it is quite easy to locate and confirm information. This makes it easier to help honest people out and it makes it easier to catch dishonest people in a lie.. it least, in my experience.
So, this brings us back to the OP.
"A Cal Poly UPD officer was stopping bicyclists for riding in areas where you're supposed to walk your bike today, and I heard him tell someone that you're required to have ID on you when riding a bicycle, because otherwise they couldn't tell for sure who you were. He even claimed that he could have arrested them for not having ID... Is this true..."
If the officer was making a lawful stop for a violation, it is a legal detention. IN this case, the violator is required to provide satisfactory identification for a citation. If they are unable or unwilling to provide satisfactory identification (the officer is not required to accept verbal assurances or personal writings), they can be legally arrested and held for a reasonable amount of time, as needed to conduct a CAL-ID (fingerprint) check, adequately verify or document their identity or be seen by a magistrate (judge) in open court.
Usually, contrary to Aluisious' opinion, it goes something like this:
OFFICER: Sir, I stopped you because you are riding your bicycle the wrong way on a one-way street. May I please see some identification.
VIOLATOR: What? I don't have to show you any ID, I'm not driving.... I'm on a bicycle!
OFFICER: Sir, I'm going to be issuing you a notice to appear for the traffic violation. I need ID from you to complete the citation. If you are unable to show ID, you will be arrested.
VIOLATOR: This is bullshit! My name is Frank Jones. You can kiss my ass if you think I'm showing you any ID!
OFFICER: Mister Jones, what is your date of birth?
VIOLATOR: What, huh? Oh, January 5th, 1985.
OFFICER: Your address?
VIOLATOR: What do you need that for?
OFFICER: (into radio mic) Records check for last of Jones, first of Frank, common spelling. D-O-B of 01-05-85. Verbal info only. (turning back to violator) Sir, I need your address for the citation.
VIOLATOR: Um... I just moved. I just live a few blocks from here.
OFFICER: Hold on. (places hand on ear, pauses, then keys mic) 10-4. Sir, do you have a license or ID card issued in California?
VIOLATOR: Yes a California driver's license, but I told you, I don't have it with me, so I can't very well give it to you. I'm on a bike, I don't need a fucking license to ride a bike. Jesus!
OFFICER: No, you said you wouldn't show your ID to me.
VIOLATOR: Huh? Well, I can't show it to you if I don't have it with me. Besides, what's the big deal here? Why can't you just give me a warning?
OFFICER: Do you know your DL number?
VIOLATOR: No, is there some LAW that says I need to know it by heart?
OFFICER: How is your name listed on your DL? Is it Francis or Frank?
VIOLATOR: Frank. Frank Jones, I already told you. Can we hurry this up?
OFFICER: Any middle name?
VIOLATOR: No. Frank Jones, that's it.
OFFICER: Dispatch is advising there is no record for the name and DOB you gave.
VIOLATOR: Well, that's your problem.
OFFICER: No, that's YOUR problem. You are under arrest for the traffic infraction and section 148 of the penal code, a misdemeanor. We'll sort this out at the station.
VIOLATOR: Okay, okay, my name is Steven Johnson. Here's my student ID card. I gave a fake name because this is total bullshit and someone told be you can't require ID from me.
OFFICER: You just turned a simple infraction cite into an arrestable offense that can land you in jail. I'll read you your rights when we get to the station. Do you have someone that can come pick up your bike and bring your CA ID or DL?
VIOLATOR: Pick up my bike, why?
OFFICER: Because you are under arrest.
VIOLATOR:
