It seems that every European kingdom considered itself to be the heir to Rome. The Holy Roman Empire considered itself a successor state, various Slavic rulers went by "Czar," the Ottomans termed themselves "Sultan of Rum," and even today Romania considers itself a successor.
Were there similar things going on elsewhere in the world?
Islamic civilizations were very influenced by Rome, both practically and ideologically. However, they also looked to the Prophet Mohamed's rule and the rule of the next four Arab Caliphs who ruled immediately after him (the Rashidun, or rightly guided Caliphs).
Unlike most religions, the founder of Islam was very much a historical man who founded an impressive empire with it's own code of laws and ethics. Muslims up to the modern day look to the governments of th Rashidun Caliphs and Mohamed as examples to follow.
Interestingly enough the Ottoman Empire saw itself as an heir both to Rome and Mohamed.
The relation between the ottomans and the roman empire seems to disturb some of you, it shouldn't if you reconsider your point of view. The byzantine empire never existed according to the byzantine sources. The byzantine state was called basileia ton romaion, kingdom of the romans. So when parts of anatolia was conquered by the muslims the sultan of this region was sultan of the counry of the romans, sultan of Rum. When Mehmet II conquered Constantinople he entitled himself kayser-i-Rum, a translation of the "byzantine" title. Here is a great article about it by Rolf Strootman (in dutch) https://www.academia.edu/5484616/Mehmet_Caesar_de_val_van_Constantinopel_1453_en_de_doorwerking_van_Grieks-Romeinse_heerserideologie_in_het_Osmaanse_Rijk