Here's why U.S. Notes and Federal Reserve Notes existed as two distinct types of paper currency, especially in regards to their connection to bank-issued notes:

Historical Context: Banking in the U.S. before the Federal Reserve (created 1913) was a messy patchwork

National Bank Notes: Federally chartered banks could issue their OWN notes, backed by U.S. Treasury Bonds they held. This aimed to make currency uniform, but in practice, thousands of different note designs were out there.
U.S. Notes ('Greenbacks'): First issued during the Civil War. Directly backed by government credit, NOT tied to specific held assets like gold or bonds.

The Rise of Federal Reserve Notes:

Addressing the Chaos: The Fed's goal was a single, reliable currency. National Bank Notes were phased OUT, replaced with Fed Notes we use today.
Two Signatures Why?: Fed Notes had to bridge the transition. They signify 1) The Fed's obligation to redeem the note, and 2) The U.S. Treasury ultimately guaranteeing its value.
Gold Standard Era: Back then, currency WAS, in theory, convertible to gold. Notes weren't "just paper", but promises tied to a (finite) underlying asset.

Why Promissory Notes Matter:

Notes as IOUs: BOTH national bank notes and early Fed notes functioned as promissory notes. The holder could, in theory, demand redemption for the promised thing (first bonds, later gold, eventually just the abstraction of "legal tender").
Trust is Key: Currency only works if people trust the issuer has the means to back up their promises. The dual signature was meant to reinforce that during a shift between banking systems.

Modern Echoes

The Phrase Lingers: While the specific 'two signature' era is long past, the concept of currency still being tied to something of value persists in some economic fringe theories.
Ending Gold Backing: When Nixon took the U.S. off the gold standard, this completely severed currency from a 'hard' asset basis. That was seismic, even if most people never handled a two-signature note.