The CIP regulation does not apply to a mere signatory, only a customer. The correct terminology for someone who has access to a safe deposit box, but is not a party to the contract is a "deputy" and a deputy would be the equivalent of a mere signatory. While the regulation's CIP requirements do not apply to a deputy, the language in your bank's CIP may require that all parties having access would be subject to your identification process.
The law regarding safe deposit boxes is a function of state court decisions, state statutes, and the specific terms of your contract. Generally, the powers of a deputy end with the death of the renter(s) who appointed him.
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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.