I proved my identity when I registered to vote, not when I became a permanent vote-by-mail voter.
If you're already a registered voter, all you need to do is fill out and mail in an application for vote-by-mail either for a single election or permanently:
https://elections.cdn.sos.ca.gov/vote-by-mail/pdf/vote-by-mail-application.pdf
You'll note that you're asked to sign it and warned that anyone other than the voter him/herself who signs it is committing perjury, but you don't have to provide any information verifying it beyond that.
And who is going to be checking all of those signatures? Short answer - NOBODY!
The problem with vote by mail is that the Board of Elections will be sending ballots to everyone on their registered voter roles. The problem with that is that in most states, those voter roles have not been cleaned up in a number of years. There are people on those roles that have moved or died and will still be getting a ballot, if they even get there and are not lost by the USPS.
Then let's say I get my vote by mail ballot. What is to prevent me from filling mine out, and while I'm at it, my wife and daughters ballots and my deceased mother in laws ballot that will arrive at my home because this is here last place of residence, and then going to the polling station and voting again? Who would know? They are not going to know at the polling station that I already filled out a ballot.
And then when thousands of ballots are coming in by mail, who is going to go through each ballot, compare signatures and check to see if they have voted in person? 10 - 15 people? How long before they get tired of comparing signatures before they throw up their hands and say screw it, they're all good or they're all bad?
Currently, when I vote, I show my ID to the poll worker, then they find my name in their list. At that point, I sign next to my registered signature which at that point is very easy for the worker to compare. At that point, they hand me my ballot. They have verified that it is me, and by signing, they verify that I have not already voted.
If I request a mail in ballot, I fill out a form, sign it and before they send me the ballot, they compare that signature to my registered signature, make a not on the records that I received a absentee ballot and send it too me. When I get it, I fill it out sign it and send it back. Then what happens is interesting. First, they file it with the other absentee ballots and then possibly NEVER EVEN OPEN IT. If the election(s) being voted on are not within the number of absentee ballots that are received, they are NEVER COUNTED! If they need to be counted, they are opened, verified against the registration, checked to ensure that I have not already voted and then and only then are counted.