Actually, a number of religions make the claim to being God's chosen people. What is interesting to note that each claim is separate and unique as a whole. Ironically, Judaism is the only religion which gets persecuted for such a claim.
I will explain a few of the other positions in other theological beliefs, but let me start with Judaism first. "Chosen people" does not refer to superiority or a higher status in any way.
In Judaism, it is believed that there are many paths to God. As such, we do not feel the need to proselytize, nor do feel that someone is 'bad' or 'wrong' for not following Judaism.
With that said: Historians have categorized that this statement has been one of the six historical reasons for antisemitism. Please note that Christianity also defines this term in relation to Judaism very differently.
What choseness means: In Judaism, it simply means that we chose to accept the 613 commandments as binding. That's it, that's all. Since we are in a contract with God, if we break contract, we are liable to harsher penalties and fines as a result.
Think of a cell phone: If you terminate your contract early what happens? You could be fined, refused any more service, or taken to court right? Well, same concept with Judaism here. Since we knowingly agreed to these 613 commandments, we are under contract to fulfill them.
Since no other nation chose this, they are not bound to these commandments. They basically get the sweeter end of the deal. They have the 7 base commandments to follow and they're labeled as righteous people. Jews on the other hand, have to fulfill a whole slew of commandments to even get a tenth of this status.
Yes, the Jewish people have been through countless prejudice and persecution. Yet, we have survived in light of it all: Can you think of another ancient culture that has done the same? ;)
Christianity:
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In Christianity, the concept is called: Supersessionism. This is the traditional Christian belief that Christian believers have replaced physical Israelites as God's chosen people.
Seventh-day Adventism:
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Adventists are "chosen" by God to proclaim the three angels' messages of Revelation 14 to the world.
Mormonism:
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In Mormonism, all Latter Day Saints are viewed as covenant, or chosen, people; they have accepted the name of Jesus Christ.
Islam:
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Islamic supersessionism presents Muslims as the only people chosen to carry the true word of God.
Brahma Kumaris:
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The Indian born Brahma Kumaris believe in a strict hierarchy of human souls in which they occupy their members are top 8, 108, 16,108 and 900,000 most spiritual human beings, the only religion which God talks to in person and the only one that will both inherit and rule Heaven on Earth for 2,500 years.
Rastafari movement:
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Rastafaris beliefs contain six fundamental principles, including the complete chosenness of the black race in the eyes of Jah (God incarnate), rendering them supreme physically and spiritually to all other people.
Unification Church:
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Reverend Moon teaches that Korea is the chosen nation, selected to serve a divine mission. Korea, Moon says, was "chosen by God to be the birthplace of the leading figure of the age" and to be the birthplace of "Heavenly Tradition" ushering in God's kingdom.