The easiest way to detect if something is real money or fake money is to check whether it can perform the primary function of money, which is to act as a means by which parties exchange goods and services in the market. Real money is capable of performing this function, while fake money is not.
Your line of thinking is some kinda wrong because you are basing your whole argument on the premise that money is real! Well, it's not, at least the USD which you allude to by referencing the Federal Reserve is not...
Yes, it's true, BTC is not backed by anything other than the faith placed squarely on it by the fools who buy it and the faith exercised by the even greater fools who buy it from those lesser fools. Yes BTC might as well be just numbers in cyberspace, a mirage. All this is true.
What is also true but less palatable to comprehend is that the U.S. dollars that are issued by the Fed that you made reference to are likewise an illusion. They too consist mainly of numbers that are also just domicile in the cyberspace. Sometimes theyre stored in paper or coins, but while the paper and coins are material, the dollars they represent are not. U.S. dollars are not backed by anything other than the faith of the fools who accept it as payment and of other fools who agree in turn to accept it as payment from them.
The main difference between BTC and the USD is that, for the moment at least, the illusion, in the case of dollars, is more widely, more fiercely believed and I might add vehemently defended by the same fools who accept it as payment and the greater fools who agree in turn to accept it as payment.