The argument is not that "the dollar" is some kind of uniquely American concept, but that there is a legal object called the US dollar. There are laws applying to that object, including that you can't make a fake version of it.
The fact that the liberty dollar happened to cost more than a real dollar doesn't make it not a counterfeit; it makes its manufacturer a fool.
The Liberty Dollar was not a "fake version" of the US Dollar. It wasn't even a "fake version" of the Federal Reserve Notes (nominally denominated in US Dollars, but trading at a far lower value) which make up the vast majority of circulating currency and which too many people incorrectly refer to as the US Dollar.
> The fact that the liberty dollar happened to cost more than a real dollar doesn't make it not a counterfeit…
No, the fact that it was never intended as a counterfeit is what makes it not a counterfeit.
The cost in materials is comparable to a real US Dollar (not to be confused with a Federal Reserve Note); the full cost could actually be lower due to technological advancements.
The fact that the liberty dollar happened to cost more than a real dollar doesn't make it not a counterfeit; it makes its manufacturer a fool.