the Wikipedia articles on Abbasid Caliphate - Wikipedia and Fatimid Caliphate - Wikipedia seem to indicate that both these dynasties claimed the office of caliph in the 12th century at the same time before Fatimids ended in 1171. However, there is no specific discussion of the issue, I guess it is left to the diligent reader to tease out.
Anyway, so what do we know about this period? If the rulers of Baghdad and Cairo both claimed to be the one and only Commander of the Faithful, what were the views of their contemporaries? What do Muslims today think about this situation in retrospect?
The Abbasids were Sunni, the Fatimids were Shiite.
The division actually started in the 10th Century - the Abbasids had held the Caliphate after the overthrow of the Ummayads in the 8th Century, but when the (Shiite) Fatimids took over Egypt they took to calling themselves Caliphs to show they weren’t subject to the Abbasids. Shortly afterwards the surviving Ummayads in Spain also claimed the title of Caliph, making a total of three.
Basically as the Islamic world evolved politically, the title “Caliph” drifted from meaning “leader of all Muslims” to “head of the ruling Muslim dynasty” to “ruler of a major Muslim state” following the break-up of Abbasid hegemony.