Basic identification should be conducted to ensure you "know" this individual sufficiently for OFAC verification (legal name) and fraud protection measures. Do you really want someone entering your vault area and placing/removing unknown items from a safe deposit if you don't know who he/she is?
Whether it rises to the level of CIP is up to the bank. IMHO it should come pretty close if not mirror CIP procedures/processes.
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Life without Jesus is like an unsharpened pencil - it has no point.