Русско-англо-русский юридический словарь

A checking account is a deposit account with a bank or other financial firm that allows the holder to make deposits and withdrawals. (Investopedia)

If you’ve had difficulty opening an account with a bank or building society, a credit union current account could be a good option for you. (MoneyHelper, moneyhelper.org.uk)

In school I learnt that "account at a bank" is American English and "account with a bank" is British English. (StackExchange)

The Google Ngram (here) shows that they are used about equally. If there is any difference, 'at the bank' might mean 'physically', as in 'I keep my jewellery at the bank', but an account is not kept physically 'at' the bank. (StackExchange)