Islam in the Ottoman Empire
Islam was the official religion of the Ottoman Empire. The highest position in Islam, "caliphate", was claimed by the sultan, after the defeat of the Mamluks which was established as Ottoman Caliphate. The Sultan was to be a devout Muslim and was given the literal authority of the Caliph. Additionally, Sunni clerics had tremendous influence over government and their authority was central to the regulation of the economy. Despite all this, the Sultan also had a right to decree, enforcing a code called Kanun (law) in Turkish. Additionally, there was a supreme clerical position called the Sheykhulislam ("Sheykh of Islam" in Arabic). Minorities, particularly Christians and Jews but also some others, were mandated to pay the jizya, the poll tax as mandated by traditional Islam.
Ottomanism
Ottomanism was a form of nationalism, likely inspired and created as a reaction to European ideas of nationalism and the growing Western involvement in the Ottoman Empire. Following the Tanzimat reforms, Ottomanism developed of a need to bring the Empire together. The Ottomans feared the growing threat the Europeans posed, especially after events like the Treaty of Balta Liman that allowed for British merchants in the Empire to be taxed equally to the locals, and the growing concern of the Great Powers over the treatment of Christians and Jews within the Empire. The Ottomans thought that if they could unite the Empire fully under one state entity, then they would be stronger and the Europeans would have a harder time encroaching on Ottoman territory, as well as Ottoman people. Previously, the empire was vastly split into many small communities that mostly governed themselves. The Sultan oversaw these communities, however most adhered to their own laws and beliefs. This was a reason the Ottoman Empire was so successful, because the Sultan didn't force any major changes on populations as he conquered them. Though because of nationalism, the rise of nation-states with shared senses of identities began to rise in Europe, most notably with the Greek War of Independence which also started affecting the Ottoman Empire. From these instances Ottomanism was developed as a social and political response, and a hope to help save the Empire from its downfall.