I must say I respectfully disagree with the answer provided by /u/Peepeepoopooman1202 for the plain reason he does not seem to fully address the question, or more importantly, the source.

Now I must also preface the Ottoman State is quite literally the breadth of a continent away from my area of (partial) expertise, and the author from the aforementioned source, Kaushik Roy, clearly has written extensively on the topic. He is however a military historian, his area of focus is not conceptions of continuity or ethno-cultural identity, which is more relevant to this, so the chances of a slight oversight is always a possibility.

Your source mentioned goes as follows

On 29 May 1453, Constantinople fell after a seven-week siege and the Byzantine Emperor died fighting.[135] Constantinople probably had the strongest fortifications in contemporary Eurasia. The capture of Constantinople proved that the Ottomans were masters of siege warfare, After the capture of Constantinople, the capital of the Byzantine Empire became the capital of the Ottoman Empire. The Osmanli Turks called their empire the Empire of Rum (Rome). [136]

This simply isn't true, at all. This is also a misconception I've seen before, bizarrely enough also in Wikipedia. As an aside, I've noticed there seems to be some person(s) on Wikipedia utterly obsessed with putting something to the effect of "The Ottomans called themselves the Roman Empire" somewhere in the Ottoman Empire wikipedia page, it's been a little edit war for some time now. I'll speak no further of it, but it was worth mentioning.

The Ottomans categorically did not consider their Empire to be the "Roman Empire" or of any remarkable "Roman" nature. They had defeated a people they knew as the "Rûm" (the Byzantines, who knew themselves as Roman) and for some centuries before the Ottomans, the Byzantine State and large swaths of culturally Greek Anatolia had generally be known to the Muslim world as Rûm. Most notably, the Seljuk Turks established an expansive State in Anatolia following their defeat of the Byzantines centuries before the rise of the Ottomans, which we know as the Sultanate of Rûm. To the Ottomans, "Rûm" referred to either the ethno-cultural group we would now simply call Orthodox Greeks, or the territories they inhabited. To say the Ottomans called their Empire the "Empire of Rûm" would be like saying the United Kingdom called itself "Ireland" after conquering the island.

I wish I could give you a more comprehensive answer, but I'm limited by the fact I do not have access to the Bibliography for Roy's book to see his citation, in addition to the citation for the other claim (Exalted State of Rome) citing a book from 1873 written in in what I assume is Ottoman in Persian script.

To understand this question we must first understand a political concept that arose through the conquests of Justinian I. This concept is known as Translatio Imperii, which refers to the idea that an empire may be built upon the structure of a previous one. In the case of the Roman Empire, the Byzantines regarded themselves as a true successor of Rome through the concept of Translatio Imperii, which meant that after conquering Constantinople, it may have been argued by the Ottoman rulers after Mehmed II, that through Translatio Imperii, the successor of the Roman Empire was indeed the Ottoman Empire.

Through the years 1523 to 1536 then Grand Vizier Ibrahim Pasha invested in scholars and research into the concept of Translatio Imperii, creating a theological and historical discourse dedicated to creating an image of Ottoman hegemony. This served a double function, on the one hand, this dicourse of hegemony could allow for the cohesion of the already diverse Ottoman Empire, and on the other hand create an international and even geopolitical ideal of the importance of the Ottoman Empire as a superpower in the Mediterranean region.

This final point regarding the importance of Translatio Imperii as a way to spread the idea of Ottoman hegemony internationally was quite important for the relations between the Ottomans and the European powers. A letter written by the Pope Pius shortly after Mehmet's conquest of Constantinople invites the Sultan to receive baptism as a Catholic in exchange of having the Vatican's support in his claim and being crowned Roman Emperor.

In short, it is not only true but an important part of both the identity of the Ottoman Empire and its subjects, but also an important aspect of the international relations agenda followed by the Ottomans during the 15th and 16th Centuries.

Sources:Writing History at the Ottoman Court: Editing the Past, Fashioning the Future - H. Erdem Cepa.Letter to Mehmed II”, in: Christian-Muslim Relations 600 - 1500 - Nancy BisahaThe Religious Concordance: Nicholas of Cusa and Christian-Muslim Dialogue - Joshua Hollman

Edit: Typo

Hi! I believe this comment from u/kedimobey, while focusing more on the title Kayser, should answer your question. https://www.reddit.com/r/AskHistorians/comments/a2kaep/when_mehmed_ii_claimed_himself_kayseri_r%C3%BBm_was/eb08vb4/?utm_source=share&utm_medium=ios_app&utm_name=iossmf