I answered a similar question about old banks here.

Proving identity was indeed sometimes a problem. Not perhaps proving you were the depositor as much as proving a note ( think "check/cheque") from a depositor was genuine. There would be a signature on file to compare with the signature on the note, which was important. There could also be a delay- if it was large, the banker might want to first have the note verified- send a query to the depositor asking if it was theirs, etc. And forgery in England carried the death penalty.

Cashing your note at another bank was also difficult, for the above reasons. But also for another: the note itself might be drawn on a bank that had failed, and so be worthless. With slow communications, there was always a danger the banker might cash the note without being aware of that. Again, he could just delay until he knew. Or he could discount the note: give you only 20 pounds for a 25 pound note, say, the rate depending on how much risk was perceived. But if you knew you were traveling from London to Birmingham, you could get your London bank to write a note to a Birmingham bank- again, for a fee- and that "letter of credit" was more easily cashed.