Japanese and Chinese Emperors never traced any roots to rome, Japan is the only nation where an Emperor is still head of state, and the Emperor is considered the most important figure in the Shinto 'religion' (if that is the right cognate)
Simularily with Vietnam, though the title was not recog ised outside of Vietnam.
Only Mongols descended from the Great Khan up until the fall of the Yuan dynasty are recognised as Emperors, again, no real relation to Rome.
Byzantium was indeed the continuation of Roman tradition, ad ruled by Roman Emperors, up until 1453, when the Ottomans claimed the title. Likewise the Czar of Russia (Czar derived from Caesar) married one of the last descendants of the Byzantium Emperors and claimed to be a third Rome, at least as some kind of spiritual successor.
The Holy Roman Empire was an attempt to revive the western Roman tradition (far less legitimate than the Eastern Roman Empire, as the HRE was neither Holy, nor Roman, nor an Empire) here again we see a connection between religion and Emperors - with the Pope crowning the Emperor to add legitimacy. However this title never managed to re-unite western Christendom.
The first French Emperor had no such problem, Napoleon simply declared himself Emperor and had a Coronation without need for the Pope, he was followed by his Nephew, who as President of the Republic organised a coup and declared hinself Emperor (which the people of France then voted in favour of, securing his position of Emperor of the French - not of France noteably)
The First British Empress was Victoria, who was annoyed that her daugther (Wife to the German Emperor at the time) had teh ically a higher rank than she did. So the Empire of Indi was declared, with Victoria as it's Empress. Thus resolving the issue. Though it is notable that term British Empire (or indeed English Empire) goes back earlier, to a rejection of the Pope's legitimacy as religious leader in England. (ie if uou don't accept the papacy, you need your own Empire...)
The Empires of Mexico and Brazil are similarily self-declared titles with no influence from Rome (or the Pope). And despite claims to the contrary, no Emperor of the United States has ever been recognised.
The Austrian Empire is kinda stupid, and mostly a result of the internal politics of Austro-Hungary, along with Austria's changing place within the German states, as the Holy Roman Empire ceased to be and Prussia became the dominant force in what was to become the brief German Empire (ruled by a Kaiser again derived from Caesar).
So many but not all uses of the word Emperor are derived from the 'August' Caesar Octavious of Rome. (Thiugh he takes the name Caesar from his adoptive father Julius Caesar. (Notable the months of August and July are still named after Julius Caesar and Caesar Augustus)
When you talk about other cultures where they have had the title of their ruler translated into English as Emperor, it becomes unclear whether the definition is being bent. But if (in english) the word meant 'those claiming to be the legitimate ruker of rome' then it would not be applied outside of Europe. And even within Europe, the title was applied in Britian ad France without such claims (and to a lesser extent in Germany).
The title of Roman Emperor passes directly down through Byzantium to 1453, by which time a western 'Emperor' is being crowned in the Holy Roman Empire by the Pope. (The Patriarch of Constantinople probably still crowned the Byzantium Empire)
Then both the Ottomans and Russia Czars claim the title. At which point it ceases to really hold this meaning. In My Humble Opinion.