I apologize for delayed response.
Before this, I was not aware that you had adopted your daughter and thereafter your husband legally adopted her as well in the UK. Before children born before 1986 SS numbers were not mandatory to clain dependents.
However, that being said, even if you are willing to obtain SS number, you do not have her birth certificate or any other birth document / hospital documents as well. Further, it is not feasible to send all the documents all the way to US.
Coming to your core question, whether banks can ask for Social Security Number or not depends upon many factors.
First, usually banks ask this for personal identification. It is NOT AT ALL mandatory. It is best adivsed to first reach out to bank in scotland and know as to WHY they are asking for SS Number. It is possible that they are asking because you might have mentioned that she is a US Citizen. Ask them as to WHY they need. IF they need it for any indentification see if you can provide them with another documents that relates to her present place of residence/ country.
Now, though it is not mandatory that banks should ask for SS Number, it still continues to remain their prerogative and a matter of internal policy. There is little we can do about them but to request them and find some way out with alternative documents.
I am sure this would help. I wish you good luck.
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