You are entitled to be informed of the person's Christian name and surname when dealing with employees of the DWP over the telephone, for the very reason you have stated.
Maybe this lady by declining to give her surname had something to hide?
Crazydiamond please - Where does it state this, in the DWP literature??? It would be so helpful to be able to quote the source of this information to any DWP staff who refuse to give their name.