Normally the number should be unique enought to identify the bank account. In spain the bank number is part of the account number so it is possible to identify the bank directly from the number. Is this not the case for your country?
I think it makes sense because people does not remember the numbers but they can remember the bank name (not the number of the bank in the account number). So the use case will be that you have to make payment but all you know for sure is that it is on the account on a specific bank.
Here's a screen capture when you have to choose bank account for SEPA payment...
Unless you really know your bank accounts (or you have few accounts), i think it make sense to have a better rec_name for accounts to allow user to find the right account easily... (maybe currency could also be included in the rec_name ?)