It is not.
Banks just have to collect more info such as Passport and national ID numbers instead.
After 9/11 I opened a US bank account without a tax ID #.
And after I got a non-resident non-citizen taxpayer ID # (ITIN) banks didn't even bother to ask for duplicate documentation. You don't need a SSN at all.
http://www.irs.gov/Individuals/Individual-Taxpayer-Identification-Number-(ITIN)
(note it can take some months to get your hands on such a number after application is sent)

But note you will need to travel to the US to do it personally, usually banks will not open an account remotely if you do not have an ITIN or SSN.