I think everyone is missing an important point. In most states, a marriage license is the legal document that "allows" you to change your name. So if the last name does not match on all the various documents (Driver's License, SS card, Marriage License, school ID, house title, credit cards, bank account etc), the marriage license is the legal document that "authorizes" a name change. So the legal name is changed once the marriage license is filed with the Vital Records (County Clerk in my state) and a certified copy is in hand. (Provided the bride wants to change her name). The OP states the marriage license is in her maiden name (of course it is). She mentions bank accounts and SS card in her post but these are meaningless. The certified copy of the marriage license is the authorizing document. Now, there will be a problem with SSA or bank if she does not take the certified document and update her accounts. In my state, you cannot have 2 legal last names; one or the other.
A court order or divorce decree are the other two common documents. So as mentioned, if you CHOOSE to change your name due to marriage, the marriage license as filed in your country, is your "proof". Make sure to have the marriage license with any forms you submit. In many states, there is a 1 or 2 year period where you can change your name with a marriage license. My nephew took his wife's last name and it was no problem although he did say some people raised an eyebrow.
I married my HS sweetheart and she took my last name. We got divorced after 25 years and she changed back to her maiden name. I soon realized it was a mistake to divorce. We started "dating" again and she soon moved back into the house under a 5 year probationary period. Yes 5 years. After my probation, we got re-married but she did not take my name because it would have been way to complicated. Most people were confused as to what her legal name was. She had 2 years to take my last name again or keep her maiden name (which she did).