Hello, I have an MA in ancient and medieval Rome, so I'll take a shot at it. It is true that Mehmet II believed that he was the new Roman Emperor by right of conquest after he captured Constantinople in 1453, but he was never acknowledged as the Roman Emperor outside his own dominion. Hence why Mehmet II wanted to capture Rome itself; it would legitimize his claim.

Keep in mind that at the time there were not less than four claimants to the title of "emperor of the Romans" including Mehmet himself. Allegedly emperor Constantine XI gave his title of Roman Emperor to the king of Castille, Leon and Aragon (modern Spain). There was also the Holy Roman Emperors in Central Europe, whose clamaints were from the House of Hapsburg. Finally the Rurik Dynasty in Muscovy also claimed the title of Roman Emperor. Ivan IV famously declared himself "Tsar of all the Russias" and declared Moscow as the "3rd Rome." It also helped that the House of Rurik had also married into the ruling Byzantine house. To complicate matters further, there were still members of the Palaiologoi family that were still alive after 1453 and the emperor of Trebizond also declared himself as Roman emperor.

Speaking of Byzantines, it was the 16th century that western scholars start referring to the Roman Empire in the east as the "Byzantine Empire" in an attempt to paint the Holy Roman Emperors as the legitimate heirs of Rome. So in short, "Roman Emperor" was mostly a title Mehmed II held for himself and his successors. No one, however, outside of the Ottoman Empire recognized his claim. Plus, the Roman Emperoro had the power of naming new Patriarchs to the Orthodox Church and no one wanted that power going to an Islamic Sultan. Even the Orthodox Church within Ottoman territory didn't recognize Mehmed II's claim, opting to declare the Roman Empire dead.