JohnM has provided some good stuff, but I would modify it a bit to say that the primary reason is his first point (which I would further modify to say “refusal to asssimilate”). The remaining points are all derived from the first.
Jews know who their God is and they are not going to either surrender him to some other imposed god or allow him to be watered down by letting him “join” someone else’s pantheon.
As you note in your question, the antipathy to the Jews is not merely a Christian problem. Jews have been oppressed by Greeks, Romans, Persians, and Arabs, to name a few, for their adherence to their God.
Since Christianity has had the longest run at oppressing the Jewish people, there are more examples of that oppression (and more excuses offered by Christians for that oppression), but it has not been an exclusively Christian practice. Roman emperors and Zoroastrian priests each tried to stamp out Judaism in their times.
Regarding JohnM’s specific points:
-Rejection of Christianity follows on the refusal of the Jewish people to change their covenant with God.
-Scapegoating has generally been a political tactic aimed at the Jews because they were convenient, not an actual belief arising spontaneously from the people.
-Money (from moneylending) was an unfortunate accident that played into the hands of scapegoaters.
Both Christians and Jews followed the prohibition against charging “usage” for money within their own religious group. So Jews could lend to Christians with interest and Christians could loan to Jews with interest, but neither group could charge interest within the group. Since there were far more Christians than Jews, the loaned money seemed to go one way and the interest charged seemed to go the other. When any king or bishop needed to absolve themselves of a debt, trumping up charges to blame on the Jews (and then confiscating their property) was a convenient method to do it.
Eventually, the scapegoating took on a generational aspect among the Christians. The Zoroastrians fell back to being a minor group in Persia in the face of outside threats. The Roman emperors swayed back and forth until Constantine decided he could do better by enlisting the Christians than persecuting them–thus ending the sporadic attempts at emperor worship. Only among the Christians was there an ongoing record of successive charges to which new scapegoaters could point as “history.” (After their initial battles, Muslims relegated Jews to a status of second-class citizens, but they did not engage in the persecutions that other groups had.)
Even within Christianity, Judaism was not persecuted uniformly. In the Byzantine Empire, early feuds between Christians and Jews dating to the first century lingered. Later, when the Byzantine Empire briefly overran portions of Zoroastrian lands, their antipathy was reinforced. In Western Europe, however, there was initially no large scale, continuous history of persecution. There were periodic calls from church leaders to avoid Jewish influence, but it was never pervasive. Charlemagne codified some anti-semitism in the West as he imported some Byzantine rules while trying to legitimize his empire. However, it was not until the 12th century Crusades that persecution in the West became an endemic problem. Money problems led to several episodes of scapegoating when the Crusades were called. Those episodes built on each other until anti-semitism was firmly entrenched in European culture at the time of the Reformation.
The “race” issue that JohnM mentioned is, again, an outgrowth of the scapegoating. European Christian and Jewish farmers are noted living peacefully side by side in what is now France, Spain, and Germany up until the period between Charlemagne and the First Crusade (or the Moorish invasion in the case of Spain). At that time some people began speaking against the Jews as a “evil race.” (This was very likely going on much earlier in the Byzantine Empire, but I have seen fewer references to it, so I am not completely sure of it.)