Any combination of 2 including birth certificate, social security card, or voter registration card can be presented. All of those documents are provided to citizens free of charge.
This is incorrect.
First, your "any combination of the two" with the examples you provided would not be proper identification for your employer to properly fill out the I9.
To fill out the I9, the employer needs an ID from either Column A or an ID from both Column B and Column C. The first two examples you gave were both from Column B, so that combination would be invalid. Reading is FUNdamental.
As per the types of ID required...
The IDs in Column A and Column B are, mostly, Photo IDs (with a few exceptions - most of which apply for individuals under the age of 18 - which wouldn't apply to voters).
With the specific examples you provided - let's talk about the Social Security Card. Has anyone here ever tried to get a new or replacement Social Security card?
https://faq.ssa.gov/link/portal/34011/34019/Article/1944/How-do-I-replace-my-Social-Security-card
First, if you're over the age of 12 and you're getting a card for the first time, you have to apply in person. In my case, my closest Social Security Card Center is roughly 45 minutes away. My closest DMV is about a block away.
Now, if you're getting a replacement card, you can fill out a form and mail it in. What's required as part of this form?
As proof of your identity, you must provide a:
●U.S. driver's license; or
●U.S. State-issued non-driver identity card; or
●U.S. passport
There are other caveats to that, but it require that you provide proof that you cannot obtain one of these items within ten work days *and* the acceptance of each of those items is subject to the whims of whatever desk jockey is approving or denying your paperwork. Since it's hard to provide proof that you can't get a state-issued ID card in ten business days, I can't imagine there's many that go this route (and are approved).
Not to mention... who still has their original birth certificate? Do all these old folks who can't get to the DMV still have their birth certificate from 70-80 years ago? *and* getting copy of your birth certificate? I can't speak for every state, but in the grand ol' state of Illinois, it *requires*, once again...
http://www.idph.state.il.us/vitalrecords/Pages/vgipi.htm
A valid government issued photo identification (ID) includes a driver’s license, a state issued photo ID or a passport.
I'm still not really seeing how requiring all these forms of ID to get a job is any less discriminatory than having to flash a Photo ID to vote.
How about a compromise - Voter ID laws pulled in line with the I9 requirements. In order to vote, you have to present an item from Column A or an item from both Column B and C.
For instance, say the account is only in the husband's name, or father's. If this is a family that trusts each other, everyone else has access so long as he lets them use a debit card, or atm card (if he tell them the PIN).
I'm not saying it's impossible - just that it seems like, in the grand scheme of things, it'd be a very minor group of people who fall in that category. As for the children in your example, those under 18 probably aren't going to be voting.

Granted, some folks might let their adult children use their bank account...
Also, because I imagine the restrictions on marriage have been in place for a long time, so people accept them as a fact of life.
If this were the case, then wouldn't folks just accept the "fact" that marriage is between a man and a woman (No, I don't believe that)? That restriction has been in place, in most parts of the US, for far longer than ID requirements.
We should not and do not accept that because it's flat out discrimination. A good majority (in most states, at least) has banded together and said "NO!" to this kind of a law because it leaves folks unable to get married to the individual they love.
*IF* requiring an ID to vote is discrimination, then requiring an ID to get married should be considered the same - but until I brought it up just now, I've *never* heard a single person complain about that...