The Gemara tells the story of Rabbi Akiva and Turnusrufus Harasha (who was the ruler of Israel in Roman times). Turnusrufus asked Rabbi Akiva why the Jews make a bris milah. After all, if G-d created man in this way, why do the Jews come and change G-d's handiwork? Rabbi Akiva asked him to bring a bundle of wheat along with rolls of freshly baked bread, as well as a bundle of flax along with beautifully crafted flax-based clothing. When the items came before them, Rabbi Akiva asked Turnusrufus, "Which of these are nicer? G-d's handiwork or the handiwork of man?"
In other words, G-d created many things with the intention that man should come and perfect them into much more useful and beautiful things.
But what was their real argument, on a deeper level; did Turnusrufus really care which is nicer? Their argument was a reflection of the fundamental difference in attitude between the Jews and the Goyim. Turnusrufus was of the view of ESAV, his forefather, that man is what he is and cannot be changed for the better. We are creatures of our instincts, and although we may be able to act with honor and restraint at times, it is only if it ultimately benefits us and our desires. We can't change our nature, we are just intelligent animals. But the Jew stands for the opposite. We were given the Torah, which helps a person rise ABOVE their nature and change their instincts and animalistic drives. A human may be part animal, but he is also part soul! The light of the Torah can actually change a person, and that is what the bris milah symbolizes. The bris symbolizes that Hashem gave us this animalistic body unperfected, with the intention that we perfect it and rise ABOVE our nature.
Why was this symbol given specifically on this organ? Perhaps because it is the sexual drive that tests a man's animalistic desires the most in this world. At the bris milah, a piece of our flesh must be painfully removed from our bodies, so too, the shmiras habris - the upholding of sexual purity throughout our lives - is the true test of whether a person is ready to rise ABOVE their animalistic desires and fulfill their purpose. Upholding the bris can feel sometimes like we're ripping a part of our hearts out! That is why both the bris and the heart are sometimes called "arel" - uncircumcised, as the pasuk says "ומלתם את ערלת לבבכם" - and you shall curcumcise the "foreskin" of your hearts. When we uphold the bris throughout our adult years, we are in effect affirming retroactively how much we value the bris - treaty - that was made between us and Hashem at 8 days old without us having had a say in the matter.
The Jew and the bris are inseparable. It is a symbol of what we stand for and of our unique place in creation.