I don't see ANY requirements in ESIGN. It's entirely optional. There is a loose "handshake" standard if you and customer customers opt for the use of e-documents, but the whole law works like a driver's license and rules of the road--if you don't want to drive then there's nothing to comply with.
With regard to electronic signatures, ESIGN says "In general...a signature, contract, or other record relating to (a) transaction may not be denied legal effect, validity, or enforceability solely because it is in electronic form." That's an instruction to federal judges, not a legal requirement imposed on the signer or the businesses with whom s/he is dealing.
That's the law.
Above and beyond the law, business partners could impose more stringent standards on each other. In particular, if the banking product/service results in assets that are sold to, or insured/guaranteed by another party, these other parties could set standards that are stricter than state or federal laws.
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...gone fishing.