As I understand it from reading King & Wilson's book on the subject, the test compared AA's DNA with that of FS's great-nephew (I don't have the book to hand) through the female line, and there was a match. Some of those doubting that AA and FS were the same person have claimed that FS's father married twice and FS and the great-nephew's mother had different mothers. However, if that were correct, this would not produce a false positive match but simply a non-match.
Personally, I'm happy to accept that AA and FS were the same person. It probably helps that I was an AA sceptic from the beginning. What I do find vastly interesting is that AA managed to fool so many people, by no means all of them credulous by nature. The arguments over her language ability are just one manifestation.
To my mind, AA's refusal to speak Russian, even if she did understand it, is damning, along with the rest of the lengthy list of facts which went against her (most notably the escape story itself, which has as many holes as a colander). And that is before we get to the DNA.
Ann