No. CIP regulations require that you obtain an identifying number prior to opening an account. It's also one of the four things you must keep for 5 years after the account is closed. However, BSA does not stipulate required contents for signature cards; i.e. you keep it wherever you keep it.
Keep in mind that the IRS is interested in TINs too and TIN certifications are often incorporated into the signature card. The TIN should appear on the same document as the customer's certification, but it does not have to be on the signature card.
In this world you must be oh so smart or oh so pleasant. Well, for years I was smart. I recommend pleasant.