As it is now, you don't need to forge anything. All you need to register for vote in California is to fill and mail a form. You can put last 4 digits of your SSN if you don't provide your state ID#. You don't need to have an address, you can just draw a map where you live. Supposedly, if you register like that (by mail) you have to show a photo ID first time you vote, however, in practice, nobody wanted to see my photo ID despite my attempts to show it.
What stops someone from filing a bunch of those with random names and random 4 digit SSNs with maps to the Skid Row or SOMA or any homeless encampment? You think they might be checking name match with the SSN? Could be, but then you can also apply for ITIN with the same random names and see if they can distinguish one from another.
If ID were to be required then you'd have to make multiple fake ids - inherently more complex and dangerous task than filling out a bunch of voter registration forms. And it could be very dangerous to show with these fake IDs at the polls where they could have government-issue scanners checking them out.