What a DNA test actually measures
The consumer tests like 23andMe, AncestryDNA, MyHeritage, FamilyTreeDNA read several hundred thousand points across your genome and compare them against reference panels of people with documented ancestry. What comes back is a statement of statistical similarity: your genome resembles the genomes of a particular population more closely than it resembles others.
For Jewish ancestry, this works unusually well, and for a specific historical reason. Persecution, expulsion, and plague reduced European Jewish populations severely in the medieval period, and the communities that survived rebuilt from a small base. Geneticists call this a bottleneck. Its consequence is that Ashkenazi Jews today share long, identical stretches of DNA with one another, enough that two Ashkenazi strangers typically appear as genetic cousins. Detection is therefore reliable in a way it is not for, say, distinguishing a Piedmontese from a Ligurian.
The landmark study remains Atzmon and Ostrer's 2010 paper in the American Journal of Human Genetics, which examined seven Jewish populations genome-wide: Iranian, Iraqi, Syrian, Italian, Turkish, Greek, and Ashkenazi, and found that each formed a distinct cluster, sharing Middle Eastern ancestry while carrying variable European and North African admixture. For the Ashkenazi, Sephardic, Italian, and Syrian groups, that European component ranged between roughly thirty and sixty percent.
Two things follow from this. Jewish populations are genetically identifiable. And they are also, unmistakably, European, which is precisely what makes the finer distinctions so difficult.