Our first introduction to Shabbat is at the conclusion of the Torah's first account of Creation of the World. Shabbat is - as we proclaim it during Friday night's Kiddush - ZEICHER L'MAASEI V'REISHIT, commemorative of Creation.
As such, one would think that Shabbat should belong to all people of the world. And in some ways, it does. Most of the world has a 7-day week. Many have some sort of Sabbath, day of rest.
Whether it is Sunday or Friday or Saturday, does not change the fact that most of the world recognizes the concept of a 7-day week. And that is remarkable because a week is the only unit of time not directly related to nature as it continued from Creation. A day is based on the rotation of the Earth on its axis.
A month is based upon the cycle of the phases of the Moon. A year is roughly based on the orbit of the Earth around the Sun. Hours, minutes, and seconds are sub- divisions of a day. Micro and milliseconds are luther subdivisions. Decades, centuries, millennia are multi- ples of a year. Only the week is different. It is not a multiple of a day. It is based on the fact that G-d created the Universe and everything in it in six days and rested on the seventh.
Not all people who use 7 day weeks believe in G-d, but that is still what a week is. It would be reasonable for the Shabbat to belong to the whole world. But it doesn't. The first ten generations from Creation until No'ach deteriorated so much and turned their backs on the Creator that He almost destroyed the entire world. When He started all over with No'ach, He promised not to bring the world to near extinction... rainbow... etc. One of the ways that G-d phrased His promise was that the seasons and DAY AND NIGHT LO YISHBOTU, will not cease or rest.
This can be interpreted on a DRASH level, that Shabbat was being withdrawn from the world, as a way of G-d distancing Himself from humans. Until B'nei Yisrael came into existence. Then Shabbat was returned, but not to all. Shabbat became the eternal sign of the covenant between G-d and the People of Israel. Whatever other peoples do, however they mark the week, it is not the same as our Shabbat.
And then, as a result of the Exodus from Egypt, Shabbat wears a second hat. So in Kiddush we also describe Shabbat as ZEICHER LITZI'AT MITZRAYIM, commemorative of the Exodus. And that aspect of Shabbat is certainly ours alone. Shabbat is the original example of Sanctity of Time. And as the first of the MIKRA'EI KODESH, it shares its Sanctity with the Jewish calendar, as well.