There are two rule sets, the Internal Revenue Code and the Bank Secrecy Act.
The Internal Revenue Code only applies if the account is interest bearing. Then, you are required to get a W8-BEN or its equivalent. Otherwise, nothing.
The Bank Secrecy Act requires you to perform CIP on all new accounts. It notes that you can accept an SSN or a number assigned by the individual's country of origin. The FFIEC examination manual indicates that non U.S. persons are customers subject to examination overview; i.e. they must be identified at account inception. Whether they are or are not higher risk will be driven by the surrounding facts, but you will always gather more due diligence information and develop a profile that is more specific than you would need for U.S. persons.
Of late, banks are increasingly saying, "No U.S. TIN, no account." It makes a great deal of sense.
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In this world you must be oh so smart or oh so pleasant. Well, for years I was smart. I recommend pleasant.