The Caesar of Rome title isn't because the sultans actually viewed themselves as a descendant of the Roman empire but rather the inheritors of their legitimacy

The Caesar of Rome title isn't because the sultans actually viewed themselves as a descendant of the Roman empire but rather the inheritors of their legitimacy to rule and is actually part of the prophecies of Islam. Thus for the Sultan to be legitimate, he must conquer Constantinople and Rome (eventually) and thus fulfill the prophecy. (Source: Empires of the Sea) The comparison between China and Turkey as descendants of the Roman Empire is a bit of a stretch. China at least uses the same language (old Chinese poems written in the Tang dynasty and even Han dynasty can still rhyme correctly and make sense if read in Cantonese), holidays, culture, etc. Turkey doesn't use a romance language or Greek.