This is the flag of an Egypt where worship of the Aten managed to survive. This is because Akhenaten, in our world known as the Heretic Pharaoh, went about his reforms in a bit more subtle way. Instead of one god replacing many gods, Akhenaten created a belief system not unlike Zoroastrianism, Hinduism or Platonic Philosophy. It is believed that above all is Aten, but he is complex and difficult to known in his highest form. Therefore, to make himself known, Aten has sent out emanations of himself that were thought of as the gods of Egyptian Mythology. The pharaoh is said to be able to commune directly with Aten and hold a special place in the new religion.
The desire to spread the word of Aten to all corners of the globe lead Egypt to focus on expansion. Even in the present day, thought their empire had since shrunken, they remain a major player of the world stage. Their closest ally is Greece, whose religion is based around the veneration of the Agathon and is largely based around the writings of Plato and Aristotle. In the past Egypt went through bouts of religious fanaticism, but today it is largely a secular society, those Atenism is still followed by most of the population. The survival of Atenism meant that the Abrahamic Faiths never happened, so the geopolitical and cultural situation is quite different from out world.
The black stands for the black fertile soil of the Nile River. Black has been considered a symbol of good fortune ever since ancient times in Egypt. The gold sun in the center of the flag is of course Aten itself.