Well, just like you cannot "duplicate" a car, but you can duplicate the CAD drawings for the car, the 35 dollar bill fiat example would mean a duplication of the actual bill.
This is the block chain, right? No actual bills. So, as soon as BTU forks, if I had $35 in BTC, then I will also have $35 BTU because now TWO blocks exist in this Universe. And the forking means that transactions will now be in different block chains.
I assume someone already realized this, and there might be a "security" feature, like for example, BTU will inspect BTC transactions? But if you have a new BTU address... will it track all the way to when it was BTC, to check then BTC transactions from the fork day to today? That seems almost impossible even with a ExaHash speed network that we have now for BTC.
By the way... I just had to research... what if I just cut in the middle a $100 bill, take one to one half to one bank and the other half to another bank? Well, it turns out you there is a centralized office that handles this, not the bank directly, and an accountant friend just shared that it seems that you must have at least 51% of the bill serial number, so that only THAT part of the bill will be replaced. If anyone turns with the other 49% it will be denied... interesting, right?