I'm surprised they didn't suggest a gun licence!. (That actually happened to someone I know - a lady in her eighties.)
It is not correct that a driving licence or passport HAVE to be provided. A lot of banks etc. ask for these because it makes their job easier, & a lot of their staff don't understand the law on this, which leaves a lot of discretion to the banks as to how they satisfy themselves that the person who wishes to open the account is who they say they are.
Assuming she is on the electoral roll, that is one way they can check, & there are plenty of others they can use if they are practical about it.
If you meet with a blank wall persist and ask to see a more senior person. If you feel they are being completely unreasonable ask for their complaints procedure and follow it. Do not accept any statement that any particular document they ask for has to be provided "because it is the law".
Have youy tried the Co-op Bank? It might be that they would be more understanding.