Answer:
Your bank, generally by including a list of acceptable identification in its board approved CIP, decides what is acceptable as identification. If military ID is on the list, it's acceptable. If it's not on the list, it's not.
While the SSA flatly says a SS card is not identification, the military makes no similar disclaimer. There are regulations that are often read to prohibit the copying of military ID, but none that say you cannot accept it as identification. (Banks are not required to copy identification, just keep a description of it.)
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