As I’ve tried to tell you before, there is no such thing as a database which tells you what the value of the USD is, or the value of gold, or the value of IBM stock, or the value of a bushel of wheat or the value of a house. They are market-determined prices, full stop. The rate for USD/JPY is a reflection of what the market believes the dollar is worth in terms of yen at a given point in time.
People try to come up with value figures to try to determine if the market is overpricing or underpricing something, but they are nothing more than estimates. You cannot make a statement like the USD is worth this amount of JPY because the US has this many dollars outstanding against thus and such level of GDP and Japan has this ratio. The price of any given instrument at any point in time doesn’t just reflect the present, it also reflects a perception of the future.
Forget about the old gold standard. That was when a dollar was defined as being worth some fixed amount of gold. There is no fixed standard like that anymore.
But for there to be an exchange rate, all the exchange rates must have been set at some point. They would have had to have been calibrated based on something. Ever since then, I agree it can be traded based upon what the trader thinks it is worth from inflation, unemployment, etc.
There has been no external reference point in the forex market since Nixon took the US off the gold standard. Prior to that for many years the USD was the only currency officially pegged to gold. Everything else traded against it (and obviously against each other). I think you need to go back to the 1930s to the last time multiple of the main currencies were pegged to gold.
In other words, the majors have been freely floating for 30+ years, and prior to that only the USD wasn’t freely floating for several decades. If everyone is on a gold standard then all of the exchange rates are effectively set and fixed based on gold reserves. With no gold pegs anymore rates are 100% determined by market forces.