The relationship between a banker and a customer (account holder of that bank) is primarily that of a debtor and a creditor, and not that of principal-agent relationship.
Secondly, when it comes to establishing banker-customer relationship as debtor-creditor relationship, there is a vital exception to the common known rules/practices of debtor-creditor relationship.
In order to simply understand this exception - you have to ponder upon only one important question i.e. "what happens when the debt is due from the borrower?".
So, let's discuss this in detail:
1) In case of a debtor and a creditor, the debtor (who owes money to the creditor) is bound to return the money whether the creditor makes a demand on him to return the money or not. This is of course as per their agreed terms on period of time, interest, installment amount, etc.
2) However, in case of a banker and its customer, the banker is only liable to pay back the money to the customer only when a demand is made in that respect. Bankers are not responsible to find the customer and make the payment. Customer is required to make necessary application to withdraw his money from his credit balance in the bank.
Case law: The following important case laws will help you in better understanding of the matter in hand.
[United Kingdom] [House of Lords] In Foley v Hill (1848) 2 HLC 28, 9 ER 1002, it was held that "Money, when paid into a bank, ceases altogether to be the money of the principal; it is by then the money of the banker, who is bound to return an equivalent by paying a similar sum to that deposited with him when he is asked for it. The money paid into a banker’s is money known by the principal to be placed there for the purpose of being under the control of the banker; it is then the banker’s money; he is known to deal with it as his own; he makes what profit of it he can, which profit he retains to himself, paying back only the principal, according to the custom of bankers in some places, or the principal and a small rate of interest, according to the custom of bankers in other places. The money placed in custody of a banker is, to all intents and purposes, the money of the banker, to do with it as he pleases; he is guilty of no breach of trust in employing it; he is not answerable to the principal if he puts it into jeopardy, if he engages in a hazardous speculation; he is not bound to keep it or deal with it as the property of his principal; but he is, of course, answerable for the amount, because he has contracted, having received that money, to repay to the principal, when demanded, a sum equivalent to that paid into his hands."
[Court of Appeal of England and Wales] In Joachimson v Swiss Bank Corporation  3 KB 110, Hon'ble Judge Atkin L.J. read the following judgment: This case raises the question whether the customer of a bank may sue the banker for the balance standing to the credit of his current account without making a previous demand upon the banker for payment. It is unnecessary to recapitulate the circumstances which raise this issue in this case. It is sufficient to state that upon the hearing of a summons for particulars on March 16, 1920, the defendants undertook to limit their causes of action to those arising on or before August 1, 1914, and that at that date no demand for the balance had been made to the bank. The question seems to turn upon the terms of the contract made between banker and customer in ordinary course of business when a current account is opened by the bank. It is said on the one hand, that it is a simple contract of loan; it is admitted that there is added, or super-added, an obligation of the bank to honour the customer’s drafts to any amount not exceeding the credit balance at any material time; but it is contended that this added obligation does not affect the main contract. The bank has borrowed the money and is under the ordinary obligation of a borrower to repay. The lender can sue for his debt whenever he pleases. I am unable to accept this contention. I think that there is only one contract made between the bank and its customer. The terms of that contract involve obligations on both sides and require careful statement. They appear upon consideration to include the following provisions. The bank undertakes to receive money and to collect bills for its customer’s account. The proceeds so received are not to be held in trust for the customer, but the bank borrows the proceeds and undertakes to repay them. The promise to repay is to repay at the branch of the bank where the account is kept, and during banking hours. It includes a promise to repay any part the amount due against the written order of the customer addressed to the bank at the branch, and as such written orders may be outstanding in the ordinary course of business for two or three days, it is a term of the contract that the bank will not cease to do business with the customer except upon reasonable notice. The customer on his part undertakes, to exercise reasonable care in executing his written orders so as not to mislead the bank or to facilitate forgery. I think it is necessarily a term of such contract that the bank is not liable to pay the customer the full amount of his balance until he demands payment from the bank at the branch at which the current account is kept. Whether he must demand it in writing it is not necessary now to determine. The result I have mentioned seems to follow from the ordinary relations banker and customer, but if it were necessary to fall upon the course of business and the custom of bankers, think that it was clearly established by undisputed evidence in this case that bankers never do make a payment to a customer in respect of a current account except upon demand."
[India] [Calcutta High Court] In Santosh Kumar and Ors. vs The King (AIR 1952 Cal 193), it was held that "The relationship between a depositor & a bank is the simple relationship of a creditor & a debtor. A depositor who deposits money in a bank in his current account is nothing more than a creditor & it cannot be said that there has been any entrustment to the bank for any particular purpose. The bank is of course liable to refund the money to the depositor when the depositor calls for it, but the money deposited belongs to the bank & the bank is entitled to deal with it as it likes.".