I just looked at my California vaccination card, vaccinated by the city of Los Angeles. There is no ID number near the CDC logo. I did get a QSR code from Carbon Health, the provider that was working with the city of Los Angeles to administer the vaccines. I also signed up online at gogetvax (google it, they do a good verification) and got a scannable code from them, you do have to trust them with some personal info there. Both scannable codes loaded to my Apple wallet, and I also keep a separate photo album “documents” in my phone for docs such at these.
None of these helped me with recent trips to Hawaii, the state had a whole separate process for PCR testing and ultimately accepting vaccination cards.