Ottoman Empire was named "دَوْلَتِ عَلِيّهٔ عُثمَانِیّه or Devlet-i Aliyye-i Osmâniyye " The Exalted Ottoman State [1]. Even though the Sultans had many titles, the empire was never called "Roman". However, to the western world it was known as the Turkish Empire [2] just like Soviet Republic being called "Russians".

Personally, I have never seen Roman in the description of the empire. However, if you are interested sultan's title looks like this:

"Sultan (given name) Han, Sovereign of The Sublime House of Osman, Sultan es Selatin (Sultan of Sultans), Khakhan (Khan of the Khans), Commander of the faithful and Successor of the Prophet of the lord of the Universe, Custodian of the Holy Cities of Mecca, Medina and Kouds (Jerusalem), Kayser-i Rum (Caesar of Rome), Padishah of The Three Cities of Istanbul (Constantinople), Edirne (Adrianople) and Bursa, and of the Cities of Châm (Damascus) and Cairo (Egypt), of all Azerbaijan, of the Magreb, of Barkah, of Kairouan, of Alep, of the Arab and Persian Iraq, of Basra, of El Hasa strip, of Raqqa, of Mosul, of Parthia, of Diyâr-ı Bekr, of Cilicia, of the provinces of Erzurum, of Sivas, of Adana, of Karaman, of Van, of Barbaria, of Habech (Abyssinia), of Tunisia, of Tripoli, of Châm (Syria), of Cyprus, of Rhodes, of Crete, of the province of Morea (Peloponnese), of Bahr-i Sefid (Mediterranean Sea), of Bahr-i Siyah (Black Sea), of Anatolia, of Rumelia (the European part of the Empire), of Bagdad, of Kurdistan, of Greece, of Turkestan, of Tartary, of Circassia, of the two regions of Kabarda, of Gorjestan (Georgia), of the steppe of Kipchaks, of the whole country of the Tatars, of Kefa (Theodosia) and of all the neighbouring regions, of Bosnia, of the City and Fort of Belgrade, of the province of Sirbistan (Serbia), with all the castles and cities, of all Arnaut, of all Eflak (Wallachia) and Bogdania (Moldavia), as well as all the dependencies and borders, and many others countries and cities."[3]

The title was taken serious as it was an official title. However, Ottomans never called their nation "Roman". Kayser-i Rum means the ruler of the Romans, so it doesn't necessarily mean they saw themselves as a direct descendant or even a cultural legacy to the Roman Empire. It was more as if they saw themselves as the successor of greatest military power in the world.

Moreover, for Mehmed the Conqueror, there was a personal meaning behind it.

Relating yourself with great figures in history was a very important matter for legitimacy and show of power/reputation. This is the main reason why in the Roman Empire the successor emperors were named Caesar Augustus. There is some evidence in the Turkish history that shows the same kind of behavior before the Ottoman empire, which leads to Mehmed II to consider himself as "Caesar".

Mohammed II (1200-1220) of Khwarazmian dynasty, a Turkic nation in the modern day Iran, gave himself the title of Iskender-i Sani [4] after defeating Kara Khitay taking Bukhara. Iskender-i Sani means Second Alexander and even though he didn't live a glorious life (defeated by Genghis Khan) he compared himself with the greatest general who ever lived.

Afdhal al-Din Abu Hamid Kermani[5] was a poet lived in the 12-13th century. In one of his poems he clearly describes the Seljuk Sultan as "Kayser":

“Kayserin ayağının altında yer eksilmekteydi

Köşkü gökyüzüne yükselmişti

Ey Keyhüsrev onun yerini almış durumdasın

Söyle o köşk nerede? Kayser ise sanki hiç yaşamadı”

So it is safe to say that the Turkish/Turkic people were somehow seeing themselves as a successor to the Roman Empire. For Mehmed II, he was a Renaissance man, he was very intrigued and studied the ancient world and classics. Moreover, he wanted to centralize the West and the East under his empire.[6] Kayser-i Rum was the perfect title to rule over the Asia Minor and Europe.

The Roman influence was still there in the times of Suleiman I (Magnificent). After the battle of Mohacs, three statues of Hercules, Apollo and Diana were erected in Constantinople which were brought from Hungary.[7] These were the first human formed statues that the common people has ever seen at that time. It caused quite controversy in the Ottoman Diwan.

Some historians speculate that, Mehmed II fought with all the dynasties who could claim the Caesar title against him. On the other hand, if the sultan wanted to expand his territories, he would have to fight in Balkans and Asia Minor which is the heart of the Eastern Roman Empire. So it is not certain, if he had this in mind. One example could be that Mehmed II sent troops to take Otranto castle in Italy (which they succeeded)[8]. However, he died shortly after and his conquest of Italy has never seen fruition.

Finally, to repopulate the city and in an effort to establish a world empire, he restored the Greek Orthodox Patriarchate and established a Jewish grand rabbi and an Armenian patriarch in the city. In the next 50 years, Constantinople became the largest city in Europe.[9] Like the Kayser title, this was an effort to create a centralized rule over the people of Eastern Roman Empire.

​

[1] Ekmeleddin İhsanoğlu (2004). Osmanlı askerlik literatürü tarihi. İslâm Tarih, Sanat ve Kültür Araştırma Merkezi. s. 741.

[2] Hamish Scott (2015). The Oxford Handbook of Early Modern European History, 1350–1750: Volume II. p. 612. ISBN 9780191020001.

[3] Özgen, Korkut. "The Ottomans History". TheOttomans.org.

[4] Michel Biran, The Empire of the Qara Khitai in Eurasian History, (Cambridge University Press, 2005), 70.

[5] Evhadüddin-i Kirmani, Rubailer (Sule Yayinlari, 2012)

[6] Halil İnalcık, An Economic and Social History of the Ottoman Empire, Cambridge, 1994

[7] TEKTAŞ, Nazım, Pargalı İbrahim Paşa, Barkod Medya, İstanbul, 2012

[8] "Pope canonises 800 Italian Ottoman victims of Otranto". BBC. BBC. Retrieved May 12,2013.

[9] https://www.britannica.com/biography/Mehmed-II-Ottoman-sultan