He had the number and his hand print.
Like the other poster said it's Swiss policy and it's called a Swiss Numbered Account. No ID necessary.
Finch: "In all the time Ive known him hes never let me down.
"Goodbye, Harold."

It's a tad poetic, but Swiss banks are/were renowned for their "ask no questions" attitude when it came to banking (this is part of the reason Switzerland will not join the EU because it would mean having to completely overhaul their banking system, which is big business to them). so while it might not be a clear cut as the movie show's it, its not far off from the truth.. also his hand print was probably guarantee enough for them.