That argument never made any sense to me. What's the fundamental value of $1?

People assign value to all things based on how desirable they are (the demand). There are rare exceptions, like when a government ties their national currency to something like the dollar, but that immediately creates a black market, where it trades for its real value, and because of that, the government exchange only works one way (to buy or to sell). See Bolivar as a current example.