It's not that I won't bank elsewhere-- when in Undercity, I often visit the Montagues, and if I happen to be in TB, I'll go to them (although, truth be told, I can't ever really remember visiting the bank in TB). On Alliance side, I don't have a specific banker that I go to, but I do like the Stormwind Bank-- it seems so much cleaner than Ironforge's bank. IF seems like a dirty pawn shop to me. Plus, Stormwind's bankers are named "Olivia," "Newton," and "John," so there's that.
The same could be asked about auctioneers, and again, on Horde I do about 80% of my AH shopping in Orgrimmar, all with the guy on the left (and I have no idea at all what his name is). There is something to be said, as a few players note, for simply going to the auctioneer that's the least busy-- it's nicer not to have to fight the lines to get a click in. And on Alliance side, there is none other: all of my auctioneer business goes through the lovely Redmuse in Ironforge. She's been running my auctions for me ever since the beginning, and I've seen no reason to change. Every once in a while, IF will run laggy, and I'll be a little irked that she's taking so long to get my auction up, but one look in those eyes and that red hair, and everything's OK again.