Judaism is a religion. If one follows the practices of that religion, one is a Jew. It has been around for so long and has such a remarkable history of resilience in the face of adversity that Jews have become also an ethnic group.
Israel is identified with the Jewish people because, of course, it was founded primary by European Jews who survived the Holocaust and migrated out of Europe after World War II. Zionism — the principle that the Jews have a homeland in the Middle East and should return there — goes back to the 19th century, and certainly laid some of the groundwork for the formation of Israel.
Not everyone who lives in Israel are Jewish. There are many Palestinians who live there, for example, as this was primarily their land before the formation of Israel, although it was occupied by the Ottoman Empire and, after its collapse, the British.
Enmity toward the Jews probably traces back to the days of the Roman Empire, as factions in Judea strongly opposed the Roman occupation of the region, and the Romans put forth quite a bit of propaganda about the Judeans at that time. If you look at antisemitic works since then, much of it can be traced, to some degree, back to this Roman anti-Judean propaganda. Likely this was in part because the Jewish people have mostly resisted acculturation through their history, and so have remained a people apart from most of the people they lived among. The same is somewhat true of the Romany people ("Gypsies"), who suffered from similar prejudice through the Middle Ages to the modern age.
Just my two cents on the subject, for what it's worth.
