Oh, god, what a mess this is.

These are the facts:

a) Your name through USCIS is 'Jones Smith, Amy' because that's what you put on the DACA form. You could have affixed an explanation of how family names are reversed in your country, typed in 'Smith Jones, Amy', and they would have given you an EAD with that name.

b) SSA has to provide you with a SSN belonging to the name that appears on your EAD, in your case it is, 'Jones Smith, Amy'. You have confirmed that your name on your SS card matches your EAD's.

c) Either as a result of user error at the DMV or SSA or a glitch in either system, SSA returned a shortened name when the DMV requested a SSN verification.

What you need to do first is go back to the SSA office and verify that your legal name is, 'Jones Smith, Amy' and not 'Jones, Amy'. If the name on their system matches the name on the card then ask them to provide you a printout. Next, go back to the DMV and inform them of the fact that your name is 'Jones Smith, Amy' and give them the printout as supporting evidence. The reason behind doing this is so that they treat 'Jones Smith' as a single last name when submitting the information for the verification of your SSN, because it's likely that they could be using 'Jones' as your last name simply out of convention and disregarding the rest of your legal name.