Under the regulation, the trust is the "customer," not the beneficiary. Nevertheless, your bank's CIP may apply to anyone you choose including the trustee and the beneficiary.
A trust is a taxable entity and must file its own income tax return. It reports income to the beneficiary on a form K-1. That form contains the beneficiary's SSN.
The trustee has the SSN.
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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.