There is no text in the Hebrew Bible that reads or states anything along the lines of circumcision being "a sign" to set Jews apart from Gentiles. Neither does Judaism or any Jew like myself teach that it was supposed to "set us apart" in such a manner.
Circumcision was a common practice of the Mesopotamian world from which the descendants of Abraham sprang. It was practiced for generations long before Abraham was born, centuries before there was a Jewish people. The rite was usually practiced upon males coming of age before going to battle or before taking a wife. It was merely received by Abraham from his culture and passed down to the Jewish world, likely without any of the ceremony mentioned in the Torah.
After the Babylonian deportation, the Jews assigned this custom a religious meaning in their liturgy by means of the Biblical texts. This religious interpretation was new, but not historical by any means.
It was not however a "sign to set them apart" from Gentiles, many of whom also practiced it. As the Genesis narrative makes clear at Genesis chapter 17:11, "it is a sign of the covenant" between God and Abraham, not a sign to differentiate between the Jews and the nations.
The Jews have been well aware since antiquity that the historicity of this sign is only symbolic in Scripture however. The same goes for much of what is written in Torah (such as our celebration of Sukkot or the Festival of Booths, which is currently going on; it did not originate during the time of the Exodus as the Bible narrative says, but comes from Mesopotamian harvest festivals in which harvesters dwelt in booths during the final days of ingathering).
Some Christians, like the Jehovah's Witnesses, as well as others have obviously added a literal interpretation to the texts of the Jews as well as misquoted details that don't even occur at all in Jewish Scripture. There is nothing anywhere in the Bible, Jewish texts, etc., that state that God gave circumcision as a sign to Jews to set them apart from the Gentiles, neither is this a part of our Jewish theology.