Employers MUST collect the SSN because SS taxes are levied and withheld against the employee's wages and needs to be processed accordingly.
Banks generally need you SSN if they need to get a credit report. If any account you have pays interest or dividends, that's taxable income. If you hold a mortgage, the interest you pay on it is tax-deductible and a frequent reason to itemize deductions (Schedule A). If taxes are involved (the latter two), the bank MUST know your SSN.
PS. If you have an SSN, yes it is your TIN. But aliens can have a TIN but not an SSN.