As a British English speaker I would say "the bank". I think it's not so much a difference between 1940s English and modern English as between 1940s banking practices and today's (especially for the more well-to-do customer) -- it implies that you have a personal relationship with an individual person at the bank who knows you and manages your money for you, and that just isn't the way banks work these days, except perhaps for the mega-rich.
Even today you "just" need qualify for slightly premium accounts - top ~2% or so will qualify at some banks - to get a named contact person at the bank you can contact, but of course at that level they still have far too many accounts per person to actively manage anything.
For my local European bank the barrier doesn't even seem to be that high only 150 k€ loans or 50 k€ investments. Then again we are not that rich country. Not that the services will be any different at that level, but hey you have named person!
I mostly agree, with the exception that I think the users of those accounts are split between people signing up for the limited extra features, and people signing up because it makes them feel important, and I suspect at least a subset of the latter would love to refer to them as "their banker". It's probably not a very large subset, though. And yes, it's pretentious.
