When using the term "Jew", it can describe someone who is ethnically Jewish (descended from one of the original Semitic tribes of Israel). The term can also refer to an adherent to one of the various faiths that identify as Judaism. It can also define someone that identifies with the culture and traditions of Judaism, but not the religious beliefs.

Lastly, closely related is someone that is not ethnically Jewish, does not practice Judaism, and does not celebrate the various traditions of Judaism. In most instances, these individuals have ancestors that were Jewish using one of the above three definition. When you ask them why they think they are Jewish, they will say something like my grandparents were Jewish.

Regarding whether Jews are exceptional. What is true is that historically due to racism, Jews were confined in many cases to ghettos and restricted with respect to what occupations they could pursue. Furthermore, frequent expulsions and pogroms along with a faith that required adherence to many strict "laws" produced a culture that emphasized education.

As the world transitioned from agriculture and the industrial revolution to the modern economy that rewards knowledge, Jews had already established a cultural value of education that once applied to secular pursuits produced an identity group that on average enjoyed superior economic outcomes.

In addition, the constant dispersion of Jews combined with a strong identity produced an international network of contacts, that served them well especially in areas of banking and foreign trade.

In other words, it is not their DNA that has made Jews exceptional, but the cultural values to some extent forced upon them, that have translated into enormous success in a modern knowledge-based economy.