Citation required. The California DMV's own website utterly contradicts what you say:
"The Department of Motor Vehicles (DMV) is permitted to issue a driver license (DL) or identification card (ID) to an applicant who submits satisfactory proof that the applicant’s presence in the United States (U.S.) is authorized under federal law...DMV will not let you start a DL/ID card application unless you present your social security number (SSN) and your valid BD (birthdate)/LP (legal presence) document...DMV will mail your photo DL/ID card after all tests and requirements have been met and USCIS has verified your legal presence status. "[1]
On a personal level, I know friends who suddenly couldn't drive because their visa expired, even though they were still in legal status while they awaited renewal. Because their license was set to expire the same day as the visa/I94. That's how hard-assed they are about it. (Btw, that's an additional non-citizen "tax" - having to renew your driver's license every 2-3 years, instead of 5 years like a citizen or permanent resident).
So definitely people without legal status can't register to vote this way. How many people with legal status do you think are going to try doing this? Like they went to all this trouble to come to the US legally, and now they'll commit a crime that has very little payoff for them personally and very high risk if they're caught? Remember these are people with a legal presence: they have jobs, USCIS knows where they live, they've been fingerprinted when applying for a visa, they're in the system. Are there FB/WhatsApp groups of non-citizens in legal status coordinating this stuff? Has anyone been arrested for this? Surely if it's going on for so long and at such a scale someone should have been caught right?
Where are your sources?
1. https://www.dmv.ca.gov/portal/driver-education-and-safety/ed...