Sultanate of Rûm

The name comes from the land in that one. Anatolia was called Rum(Rome) by the Arabs, the name was adopted by the Turks. The state or the leaders did not see themselves as Romans but as Oghuz Turks/Native Anatolians/Seljuks/Muslims.

Ottoman Sultans calling themselves "Kayser-i Rûm"

The title was taken by Mehmet the Conqueror(although i think it was also used previously) who fashioned himself as an eastern Alexander and a righteous "Asian" taking revenge on the great-grandchildren of "Moreans, Thessalonians and Macedonians" who had wronged his ancestors/friends. He did not purport himself or his empire to be Roman(as proven by his many quotes you can find online).

However in many ways the Ottoman Empire took on the legacy and systems of the Byzantine Empire before it. On that note Russia being the "Third Rome" is much more ridiculous a concept than the Ottomans Empire being the "Third Rome", as the Ottoman systems much more closely resembled the Byzantine Roman systems than the Rus state ever did. One can make the argument from religion at this point, but the Empire had already changed its religion once so it doesn't amount to much.

It should also be added here as a sidenote for the unwary that Mehmet had Byzantine blood in him and wanted to first solve the conflict in Constantinople peacefully(he offered the Byzantine Emperor lands in Greece, he offered safety to all citizens and nobles of the city, he offered the Emperor kingship and a treaty of friendship which the Byzantine Emperor declined on religious grounds stating that "No one in Constantinople has the power to hand it over to the Turks")