The Caliph (should) be from Mohammad's tribe, the Quraysh. The original caliphates- Rashidun, Umayyad, and Abbasid all could trace their lineage to that tribe.

The actual Caliph lost political power around 900. Secular power was devolved to a series of Sultans, many of them Turkic. They were content to have the legitimacy of having the Abbasid Caliphs around, granting them some legitimacy.

When the Mongols destroyed Baghdad, the Mamluk sultanate supported a member of the Abbasid family as a Caliph. By this point, any claims to secular power were hundreds of years in the past. When the Ottomans conquered the Mamluks, they took a bunch of caliphal relics and the title, but didn't claim the title for a long time, because no one took it seriously.

In the 19th century The Muslim world was under siege. Christian European powers were occupying Muslim powers left and right. The Ottomans were quickly becoming the last sovereign Muslim state in the world. At this point they dusted off the title of Caliph to grab some of the legitimacy it conferred, and sort of take up the banner of the Islamic world. While no one took them particularly seriously when they first grabbed the title in the 16th century, to the point that they stopped using it, the Muslim world was in a desperate enough state that most were sort of ok with it.

The Ottomans came up with some pretty tortured logic to claim descent from the Prophet. Throughout the 19th century Ottoman claims to the caliphate became more pronounced, and people seemed to be fine with them claiming it, as they had some decent historical claims and were the last Sunni Muslim power that could stand up to European Christian imperialism. The title had been somewhat honorific for almost a thousand years anyways.

TL;DR the title of Caliph has been pretty indefinite since the 900's. The Ottomans took the title in the 1500's to yawns. They really started claiming it in the 1800's, when they manufactured lineages that held up if you didn't look to close. The Muslim world was in dire straights, and so no one looked too close. It was better to have a symbol of Muslim power than to debate the theology.