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I've been through this with my attorney. There is nothing under the law to stop someone from using it as an identifier.
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I was responding to your assertion that your bank is using your SS # in violation of the law. I seriously doubt they are since you signing up for an account is most likely you consenting to them using your SS #.

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However, if I refuse to give them my SSN, they can't deny service.
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I doubt that - unless of course, there is a fundemental right to transact business at a given bank. If you refuse to comply with the given business practices of a bank, they can refuse to accept you as a customer. As a consumer, you have the right to choose the bank that conforms to your wishes.

On a related matter, since interest on a bank account is taxable, I don't see the unreasonableness in a bank having access to your SS #.