Yes, mostly the US grown hops of the same name are the same, old world, variety, from cuttings brought over. US Tettnang is the exception, it is indeed actually Fuggle; apparently there was a mixup somewhere along the line in passing on cuttings. No one really knows what happened, but the poor thing took a DNA test and found out it was living a lie.
But it is also true that the same hop grown on different soil will exhibit differences, from subtle to profound. For this reason there have been breeding programs to try to develop new hops for American growers that would have a character more like their old world models. Attempts to reproduce Hallertau Mittelfrüh led to sisters Crystal, Liberty, and Mt Hood, as well as Vanguard. Of these, Liberty and to some degree Vanguard are fairly close to MF, while Mt Hood actually more closely resembles German Hersbrucker. Crystal is just a rebel, the wild child of the family. Attempts to breed substitutes for Saaz have led to Sterling and Ultra. Some US grown Saaz has a nice, noble character of its own, I find, not quite Czech Saaz, but makes a tasty lager.
It's worth trying a variety of these hops, US and European, to see what you like. You may find you really prefer the originals, or not. Possible advantage of US grown hops is that they may reach us in better condition, or simply be more available.