When my wife and I married, she changed her name to [Her First Name] [Her Maiden Name] [My Last Name], like from
First: Jane
Middle: Ann
Last: Smith
to First: Jane
Middle: Smith
Last: Mylastname
All was well and good until very recently when I was at the DMV with her and we were renewing her drivers license. We found out then that the person entering her name change form at the Social Security department had misentered it as First: Jane
Middle: [none]
Last: Smith Mylastname (no hyphen, just a space)
For fun, her US passport shows it correctly, like: Given names: Jane Smith
Last: Mylastname
So two federal agencies have her name in two different ways. Yay! The DMV lady was unhappy with this but we talked her into accepting the truth on her passport so we could renew her license, but obviously you can't count on the cheerful disposition of all future DMV clerks. The correct long term answer is that we have to have her name changed legally, which will cost about $400 all told. My favorite part is that we have to run an official notice ad in the local newspaper, but that's just a plain templated text message that will read:"Notice is given that Jane Smith Mylastname is changing her name to Jane Smith Mylastname"
for which privilege we get to pay $75.
Good grief.
Source: I pretty much ran into the exact same thing.