To participate in some realestate auctions(confirmed AM, BP) you must first pay a deposit fee for the property in question.
The syntax is something like 'pay deposit on A' and the size of the deposit fee seems to loosely correlate with the size in number of rooms of the property up for auction. After the auction is over, that deposit fee is refunded to the bidder.
As far as I know, there is no way to explicitly select, with any in-game command, to which bank the auction deposit fee is refunded.
There is currently no reason to believe that the bank to which your auction deposit is refunded has anything to do with the bank selected for rent with a property rental agreement. (accessible with the 'change' command when a rental agreement is in inventory)
There is reason to believe that auction deposit refunds are deposited into the most-recently opened bank account on the continent in question, though possibly defaulting first to the NPC bank when you already have multiple accounts on that continent, and afterward to the most recent newly opened account.
More testing with young alts newly entering the banking and realestate markets might help to get a better sense of the bank selection priorities.
For example: From AM realestate deposits I was having refunds go to People's Bank of Klatch (most recently opened account), then opened an account at Bing's to test if the receiving instutution would change, and the next auction refund indeed went to Bing's. The account at Bing's was then closed until after i can effect another auction deposit refund to see whether the deposit refund reverts to PBoK or goes to some other different bank next.
This is a story but it might not reflect what is actually happens. Counterexample: In BP, my refunds go to First Imperial Bank, the npc bank on the counterweight continent, though I had since opened an account at N&L.
~~InABun, September 2021