My wife and I have SSNs as previous non-immigrant residents of the US. My wife may have had work authorization for a while in the 1990's but spent most of our time in the US without it. While living there both of us provided the SSNs on tax returns, when applying for drivers licenses (New York, Virginia and California), to my employers' health insurers, when opening bank accounts and applying for credit cards, and pretty much any other time we were required to provide a number. While non-working my wife opened her own credit accounts and had a slightly better credit score than I did. We still file non-resident tax returns using the same SSNs, and still have bank accounts in the US that occasionally ask for the numbers on W-8BENs (a Canadian bank also made us fill out W-8BENs), despite the fact we currently have no status there at all now. I also don't think my wife has seen her physical social security card for many years, but no one has ever asked to see it either. Just knowing the number is enough.
To answer the title question, then, while you can't get an ITIN, because you already have a SSN, you also don't need an ITIN, because you already have a SSN. There is no constraint against using the SSN you have already for anything that someone without work authorization would need a SSN for; for tax purposes, as you already know, you are required to use the SSN, and it is useful for credit purposes. The sole difference between your situation and my wife's is the name issue: my wife still uses the name her SSN was issued with while you changed yours, so your name is mismatched with the SSA record. Any problems that arise will be associated with the mismatching name.
Unfortunately, there is some chance the name mismatch will eventually cause you problems. For example while, fortunately, California does not require a drivers license applicant to have a SSN if their immigration status doesn't allow them to obtain one, as is true of you in your current status, they do require those who have been issued one to provide it for name verification purposes which is a catch 22 in your case, so there might be an issue if you provide the SSN before the name mismatch is resolved (I won't mention the obvious workaround, though the implication in your post that you already have a California learner's permit might suggest you are okay anyway; since you'll be renewing the license frequently and in person as a non-immigrant you will have many opportunities to provide the SSN once the name is fixed). Also credit providers, including utility companies, often submit your name and SSN to the SSA for verification, which may fail due to the name mismatch; your husband can apply for these instead but it would be better if you could do so independently if you need to. And you'll be required to file (joint) tax returns using the SSN but the IRS may kick out returns where there's a name mismatch with the SSN (they certainly will for the primary taxpayer, I'm not sure about the spouse).
So there would be some benefit in continuing to pursue the issue of the name change in the SSA files (you don't need a new card, which they won't give you anyway, you just need the name in their file changed), but I have no idea how one would go about this. What astounds me is that the SSA's own FAQ entry about name changes seems to explicitly allow changes in cases like your's: if you follow the Learn what documents you need link and select Adult-Corrected-Noncitizen it says
However, if we assigned you a Social Security number for non-work purposes, special rules apply. If DHS did not authorize you, we will change your name on our records but we cannot issue you a corrected card.
which is exactly what you need done. I do note that the application you need to fill out for the name change says
- If you check “Legal Alien Not Allowed to Work” or “Other,” you must provide a document from a U.S. Federal, State, or local government agency that explains why you need a Social Security number and that you meet all the requirements for the government benefit. NOTE: Most agencies do not require that you have a Social Security number. Contact us to see if your reason qualifies for a Social Security number.
so if you could find or obtain an IRS document stating that you'll need to use your already-assigned SSN for the tax returns they'll obligate you to file the SSA might be more cooperative(?) if it is clear you only want them to change the name in their records and don't need a new card. Unfortunately, the SSA and IRS are top competition in the "government agency that sucks the most" contest (the California Franchise Tax Board ranks highly too) so I don't expect this to be easy.
If you can get the name associated with your SSN updated my experience is that you'll be good to go, just using the SSN whenever a number is required. I wish I could be more helpful about how to go about persuading the SSA to change the name.