Character/Inventory screen: No
Bankers being able to catalog the contents of other banks: Perhaps
The old trade brokers already used the locality concept to show what's in other markets and this would be similar principles.
I'm in the No camp overall though because asset management is the personal responsibility of the character - unaffiliated banks don't really owe you any kind of record keeping service, especially since we pay zero banking fees. brokers on the other hand have a vested financial interest in people finding what they want, or placing goods for sale that they can see there is a gap in the market for.
It's these little mundane things that will help get people circulating, taking risks, making choices. Do I have x in Vadda? Am I confident enough that I do have x that I should take the time and risk to travel there?
I know I've had times where I was certain I had particular gear or resources in a town only to get there and be very disappointed, but that's all part of the fun! This should also be less of an issue with a single char game.
make spreadsheets, screenshots, commit things to memory or even the unthinkable; travel to a bank to check what you deposited there. Call me a hardcore bank patron but I would even be open to NPC banks charging holding fees for items, taking a 6 month hiatus and coming back to the game to find the bank defaulted me due to lack of payment and put all my crap on the trade broker to cover their costs. The people of Nave shouldn't put their livelihoods on hold because I got tired of living there!