I doubt whether firm could be used.
It would usually be a business organisation or company. If it has plc or ltd (UK terms) after the name (e.g. Lloyds Banking Group plc) then it is definitely a company.
'Firm' is commonly used for any company, but more strictly a firm is a limited partnership, like a firm of solicitors or of accountants, where each partner is a shareholder and it is not publicly owned.
Just to add that Lloyds Banking Group plc is not a bank but the company that directs (or has ownership of) the numerous subsidiary companies of the group, the most well-known of which is probably "Lloyds Bank" which is a bank.
A bank, as normally used, is a local office or branch of a company. The term bank does not mean a company per se.
The Bank of England, for example, may well be a company, but it is not a bank in the ordinary sense of the word.