would a law firm who represents a bank use the bank''s bates numbering system on their own internal documents? Would that mean they are one corporation

It is likely that the documents were prepared by the bank's in-house litigation staff and provided to the law firm for use in representing the bank. Generally speaking, banks and law firms are separate entities. However, no law prevents a bank from employing lawyers (as full-time employees) and having what amounts to an in-house law firm. Without more specifics, it is impossible to tell what might be going on here.