Simple join between 2 tables with unusual structure
You should probably fire the Database Admin.

But seriously, a problem in your query arises if you have duplicate, but different (bank_id + branch_id)'s. For example, (22 + 202) and (2 + 2202) both equal the same thing.

Change your query to include the space, instead of replacing it on both sides.

Here's a SQL Fiddle demonstrating the issue:
http://sqlfiddle.com/#!18/3a6b9/2

Example query to match exactly, don't remove the spaces (Note won't work on SQL fiddle as I used different lengths for the colunmns):

SELECT BANK_ID, BRANCH_ID, ACCOUNT_NUM, CURRENCY, AMOUNT
FROM [ACCOUNTS]  
  JOIN  [AMOUNTS]  
  ON (
    CAST(
         CAST  ([BANK_ID] as  nchar(2)) + ' '
       + CAST  ([BRANCH_ID] as  nchar(3)) + ' '
       + CAST  ([ACCOUNT_NUM] as  nchar(4)) 
        AS varchar(255)
        ) 
         LIKE
        CAST ([AMOUNTS].[ACCOUNT_REC] AS varchar(255))
     )


Update, since you mentioned more about your schema: 

https://dbfiddle.uk/?rdbms=sqlserver_2017&amp;fiddle=f8c037b9cc9afc2e2de850011031020d

SELECT  * FROM ACCOUNTS JOIN AMOUNTS ON (
         CONCAT(CONVERT(VARCHAR(255), [BANK_ID]), ' ')
       + CONCAT(CONVERT(VARCHAR(255), [BRANCH_ID]), ' ')
       + CONVERT(VARCHAR(255), [ACCOUNT_NUM])
       ) LIKE ([AMOUNTS].[ACCOUNT_REC] )