What requirements must a company meet to legally call itself a bank in the UK?

In the UK, you would need to be regulated by the Financial Services Authority. FSA's Official website can be accessed at: Financial Services Authority

FSA's website lists areas for which you need to be licensed:

What are the regulated activities?
Specified activities are defined in Part II of the RAO and comprise:

• accepting deposits;
• issuing e-money;
• effecting or carrying out contracts of insurance as principal;
• dealing in investments (as principal or agent);
• arranging deals in investments;
• arranging home finance activities;
• operating a multilateral trading facility;
• managing investments;
• assisting in the administration and performance of a contract of insurance;
• safeguarding and administering investments;
• sending dematerialised instructions;
• establishing etc collective investment schemes;
• establishing etc stakeholder pension schemes;
• providing basic advice on stakeholder products;
• advising on investments;
• advising on home finance activities;
• Lloyd's market activities;
• entering funeral plan contracts;
• entering into a home finance activity;
• administering a home finance activity;
• agreeing to do most of the above activities.

Specified investments are defined in Part III of the RAO and comprise:

• deposits;
• electronic money;
• rights under a contract of insurance;
• shares etc;
• instruments creating or acknowledging indebtedness;
• sukuk (shariah compliant debt instruments)
  
• government and public securities;
• instruments giving entitlement to investments;
• certificates representing certain securities;
• units in a collective investment scheme;
• rights under a stakeholder pension scheme;
• rights under personal pension scheme;
• options;
• futures;
• contracts for differences;
• Lloyd's syndicate capacity and syndicate membership;
• rights under funeral plan contracts;
• rights under regulated mortgage contracts;
• rights under a home reversion plan;
• rights under a home purchase plan;
• Rights to or interests in anything that is a specified investment listed, excluding 'Rights under regulated mortgage contracts', 'Rights under regulated home reversion plans' and Rights under regulated home purchase plans'.

Source: Do I need to be authorised?