I'm not quite sure how you come to the conclusion that Judaism and Christianity are "relatively" the same religion. They have similar roots and back stories, but Jesus literally says "there is no way to the father but through me." Jews rejected that, so they weren't Christians. Catholic tradition and teaching is almost completely based around the New Testament, with the "old law," being replaced with the laws of "Love God and love your neighbor as yourself." So a rejection of the new testament makes you not even remotely "christian."

Firstly, as TheBulgarSlayer pointed out, they were not at all the same religion in the Middle Ages, and haven't been viewed as the same by either Jews or Christians since the 2nd century at the latest.

Secondly, Protestantism and Catholicism are far more similar to each other than Judaism and Christianity, but that didn't stop Protestant and Catholic rulers from oppressing those of a different sect. And then there's the medieval suppression of "heresy", which might include people who agreed with 99% of Catholic doctrine. Indeed, one might ask why, considering that the Catholic Church declared a crusade on the Cathars, a heretical group that had split off from Catholicism, Jews were allowed to practice their religion at all in medieval Europe!

Oppressive rulers have often oppressed religious groups with what might seem to outsiders to be very small differences. Consider the Byzantines oppression of Monophysites. Or, for a more recent and vicious example, ISIS's fanatical hatred of Shi'ites. One could ask why ISIS wants to wipe out all Shi'ites: after all, aren't they all Muslims? But to ISIS, Shi'ites are more dangerous and threatening than Christians.

Now, back to the question of why Jews were oppressed. As I've said, they were (and are) in fact, substantially different religions, in practice, history, and theology, despite sharing half (but only half!) of the same religious book. But there are three other reasons for the oppression of Jews in the Middle Ages.

Firstly, they were a visible minority. Ethnically, religiously, sometimes linguistically (Yiddish), and even in their food and clothing, they stood out. Thus, they were an easy target for blame. Just as immigrants today are often blamed for disease, crime and so forth, Jews were blamed for famines, child mortality, losses in war, etc. As late-Middle Age kings engaged in the process of nation-building, they didn't fit with ideas of what it meant to be "English" or "French".

Secondly, due to being banned from owning land, and being outside the Pope's ban on money-lending, some successful Jews became money-lenders. Of course, lending money to aristocrats is dangerous, because they may not want to pay it back. King Edward of England kicked out all the Jews partly to gain favour with aristocrats who'd borrowed from Jewish money-lenders.

Thirdly, oppressing Jews was generally easy. As an unpopular group, dispersed across Europe, they rarely had the numbers to defend themselves, and lacking aristocratic commanders, they fielded no armies. Thus, they made a tempting target for Kings looking to seize wealth without upsetting the population.