Why did Egypt and Britain both claim to rule Sudan?

Because Sudan had been part of Egypt, and in 1899 Britain forced Egypt's khedive, Abbas II, to turn it into a 'condominium', a territory whose sovereignty was legally shared by both powers. British and Egyptian troops had just defeated the Mahdist state together. In practice, though, Britain ran Sudan through its Governor-General in Khartoum, and Egyptian control kept shrinking for most of the condominium's existence.

Egypt's claim went back much further. Muhammad Ali, who took control of Egypt in 1805 and ruled it almost independently of the Ottoman Empire, annexed territory in Sudan and governed it as an integral part of Egypt. His grandson Isma'il Pasha pushed Egyptian control south to the Great Lakes, hoping to build an African empire that could hold back European expansion. A ruinous defeat by Ethiopia worsened Egypt's financial problems, and the Great Powers deposed Isma'il in 1879.

In 1881, revolts broke out against his son Tewfik in Egypt and in Sudan. Britain crushed the Egyptian one, but the Mahdist Revolt took over Sudan, and in 1885 Khartoum fell and the British general Charles Gordon was killed. Between 1885 and 1898, Sudan's population collapsed from about eight million to three million amid war, famine, disease and persecution. A joint Egyptian-British force under Kitchener destroyed the Mahdist state by 1898, and Britain used that military leverage to impose the condominium.

The ending was unusual. After the 1952 revolution, Egypt's new leaders, Muhammad Naguib and Gamal Abdel Nasser, reasoned that Britain's claim to Sudan depended in theory on Egyptian sovereignty, so Egypt itself should abandon it, leaving Britain no option but to leave. Under a 1953 agreement between the two countries, Sudan became independent on 1 January 1956.