Judaism is an ethnicity as well as a religion and can be identified from DNA samples thanks to certain marker genes kept concentrated by the tendency for Jews to marry other Jews. Often, being Jewish transcends national origin simply because in so many countries anti-semitism has isolated those who are practicing Jews from the rest of the population. So yes, he could be half Jewish, just as someone else could be half-Irish.
As to his father’s parents being from Turkey, please understand that there are Jews all over the world. Most of the Turkish Jews got there from Spain in 1492 when Ferdinand and Isabella expelled the Jews from Spain (well, they could convert to Christianity but that wasn’t always the wisest choice as shown by the Inquisition). The Sephardic Jews ( the ones expelled) were welcomed by the Ottoman Empire, aka today as Turkey. There are only fewer than 30,000 Jews in Turkey today, but they do exist.
And finally, yes, you are considered fully Jewish if your mother is Jewish, but not if only your father is Jewish. However, if you are raised as an observant Jew you are generally accepted as a Jew. My cousin is Jewish in that sense. His father was Jewish, his mother was not, but he chose to attend synagogue and undergo a Bar Mitzvah so he is considered Jewish. Of course, he isn’t a member of an orthodox temple, but still he is considered Jewish by his rabbi.