Bibi is an honorific not a surname. Benazir Bibi means something like Miss Benazir.
So for a person to be called Mrs Bibi is a nonsense. Somewhere back down the line an uncomprehending official made a silly mistake.
Why does our neighbour stand for it? I've been researching Muslim surnames all morning and getting more and more confused. There seem to be no fixed conventions. If I understand my sources correctly, the purest tradition- the one that goes back to the prophet- is for everyone- male and female- to have a personal name followed by the name of his or her father. It's a system of labelling that guards against inadvertent incest- and very sensible too. Ahmed's son is called Muktar Ahmed and Muktar's son is called Hanif Muktar and so on. But in reality things are far more complicated- and practice varies from place to place and clan to clan. One thing that emerges fairly clearly is that the surname as westerners understand it- a badge of identity, honour and pride that passes down the family from generation to generation- just doesn't exist in Islamic cultures.
And I suppose that's why my neighbour isn't bothered. You want two names to put on your form? Here they are- Benazir Bibi. And if you choose to interpret the second as a surname who cares?