Short answer: ethnicity and language (both in place well before the 19th century), plus some negative experiences with other nationalities (mostly in the 19th-20th centuries, triggering the rise of Turkish nationalism.) When you read the memoirs of the future political and military leaders of Turkey, at one point almost all of them mention a story like this:

The young X (the future leader) enters an elite Ottoman school/military academy. There he meets others from different parts of the Empire. He witnesses how Albanians, Arabs (insert any other nationality here) create societies and promote their national culture. Yet he does not belong to any of these groups, and often he insists on the shared Ottoman identity. He feels alienated. At some stage, he thinks this does not work anymore (1912 Albanian independence and the Arabic revolt during WW1 being primary turning points for many, apparently) and he becomes more Turkish than Ottoman. This general story is repeated so many times that it is fair to say the Turkish identity resurfaced in the late 19th-early 20th century as a negative reaction to other nationalisms.

What Turkish culture is probably a question too deep for me to delve into here. The native tongues seem to have played a vital role in reinforcing unity among ethnic groups in the Empire, so Turkish as your native tongue (which often even though not always meant you are ethnically Turkish) plus undergoing experiences similar to the story I told made people Turks. A reminder though: it is simply a mistake to suppose that such individuals and modern states created the national identity of Turkishness. There are several much older occasions in which people spoke of themselves as "Turks" and clearly regarded others as non-Turkic/Turkish "others." A 15th century Ottoman chronicle by Mehmed Neşrî Efendi, for example, describes how Murad I of the Ottoman Empire said he ached to showcase “the Turkish manliness” against the Serbian Army, in the 14th century! So what we are talking here about is the Turkishness as understood from the late 19th century onwards but “Turkishness” itself has deeper roots in history.

Finally, I do not think that most ordinary Turks felt failed or ignored by the Ottomans. They did not really think of the Ottomans as someone else either, it seems. There was a time following the foundation of the Republic of Turkey in 1923 when the state sought to instill such a mentality (now largely exaggerated for modern political purposes). Indeed, there are speeches by Mustafa Kemal Atatürk in which he criticizes how the Ottoman conquests were merely for the benefit of the Palace and the Sultan but at the cost of much Turkish blood. The Ottoman past was often used as a yardstick to demonstrate the success of the Republic, one success being the more clear expression of Turkishness in the Republican era. But at the end of the day, I highly doubt even the elites regarded the Ottoman Empire as an enemy per se. The confrontation with the Ottoman past in the Republic of Turkey never reached levels witnessed in the French Revolution or in the Soviet Union. It was never an issue of nationality alone anyway since the Ottoman past was often criticized also for not being secular, modern and independent enough too.

Religion.

The Ottomans thought in a very simple paradigm: in the minds of the Ottoman bureaucracy they themselves were Ottoman, their Muslim subjects were Arabs or Turk, and their Christian subjects were Rûm. In the Ottoman paradigm a Muslim Roman was an oxymoron, even if he was ethnically a Greek. If an ethnic Greek was Muslim, he wasn't a Greek at all, he was a Turk who spoke Romeyka (the Roman language). To the Sublime Porte, all Christians, from the Phanariote beareaucrat to the Romanian peasant to the Kaiser of Germany and Queen Victoria were Rûm.

Mustafa Kemal was both Muslim and ethnically Turk so his identity was pretty easy.