it's not as vague as you think. Ashkenazi Jews literally share a lot of genetic material, for example, when two Ashkenazi Jews get married and want to have a child, they get tested for genetic diseases that only Ashkenazi Jews have.

And a lot of people (people who think they are Jews and also people who think they are non-Jews) are genetically mixed Ashkenazi with non-Ashkenazi.

And Ashkenazi happens one of the "tightest" ethnicities in the connected part of the modern world (Europe, America), since it is historically-long-insular low-population subgroup living within a large population.

Look up the definition of ethnicity.

I think you mean "race."

no, I don't mean race.

An ethnic group, or an ethnicity, is a category of people who identify with each other based on similarities such as common ancestry, language, society, culture or nation

common ancestry - genes.

I would dispute the notion of common ancestry is synonymous with common transfer of genes. The former is about descent, within a system of kinship, that may or may not recognize within it's domain an actual realized transfer of genes on the one hand, and may include in it's domain instances when no such transfer might have occurred (adoption/cuckoldry/politically motivated claims of kinship/etc.etc.).

come on, you are engaging in pointless sophistry. Common ancestry doesn't mean that it involves 100% guaranteed common transfer of genes every time - it just means that common transfer of genes was quite likely to take place at least relatively often. In the case of Ashkenazi Jews, we know it did.

I'm just trying to point out that you don't have to share genes with an ethnic group to belong to it.

this is not about "have to" or "don't have to", but about likelihood. Likelihood is high enough that they do genetic testing on ashkenazi jews.