So religions claiming there is a God would fall in what category?
The belief there is a God seems neither falsifiable nor verifiable. I would think claims fall in the category of doctrines adhered to by the followers of a religion, not per se the religion itself.
I would say that every religion has claims that are central to it and it would be hard to imagine that religion without that claims. If one of those claims is falsified, the religion in its current form is falsified. Of course you can then rescue the religion by dropping that claim - the same way you can rescue a falsified scientific theory by amending it. Do that often enough and you end up with a religion that is not falsifiable anymore, but might only be a mere shadow of what it once was.
YECs tend to define their interpretation of Genesis to be a central claim of Christianity. Thus they make their version of Christianity falsifiable. That does have no impact on all other versions of Christianity, unless they can convince everybody else of the centrality of Genesis. This is a major danger to Christianity, because then from the consensus that the claims of YECs have been falsified the consensus would emerge that Christianity itself has been falsified.