That said I think there are too many differences between the two to say they are truly successors to Rome.
Probably so. But whenever someone tries to put a finger on what those differences are, they melt away. So, I'm just playing with the idea; not that serious about it. This is a good forum for it as long as people don't get bent out of shape.
They didn't have the same religion …
Shrug. The emperors who sat in Rome were of a different religion (mystery religion) than the emperors who sat in Constantinople (Christianity). I don't see how that unites the two while Islam stands apart. There is some evidence Rome considered adding Jesus to the Pantheon. They likely would have considered Mohamed as well … actually would have preferred him. Dr. Ryan Reeves contends the reason Jesus wasn't added to the Pantheon was the pacifist/peace/love aspects of his preaching. The Romans considered it weak and inferior. They admired gods who fought for their people - more of an Islamic mindset.
… their people didn't even start with Romans/Greeks but Turks.
Apparently you're not familiar with some of the more extreme pan-Turkic movements, which claim Turks are the fathers of all nations … OK, even I don't take that one seriously.
But your point doesn't work for 4 reasons:
First, lumping the Greeks in with the Romans doesn't work. They are two distinct cultures. And as I pointed out, the Romans ruled many ethnicities: Slavs, Africans, Celts, Hebrews (I assume you knew that last one). Further, they provided a path for those ethnics to attain Roman citizenship (such as Paul). So, Rome itself allowed that not all who bore Roman citizenship had to spring from a parentage of the Italian peninsula.
Second, the Turks took care of the bloodline issue. Orhan Gazi married two Byzantine princesses: Nilüfer Hatun (daughter of the ruler of Bilicek) and Theodora Kantakouzene (daughter of emperor John VI). Those marriages produced sons, one of which (Murad I) was in the direct line of Mehmet the Conqueror, who took Constantinople. And there were likely others along the way. Many of the houses of Europe took power via those same tactics. No historian claims that when William the Conqueror took the throne of England it ceased to be England and became West Normandy. It was simply a new house ruling the same nation. If that were consistently applied, it seems the Ottomans would simply be a new house ruling the empire.
And that leads directly to the 3rd reason. When Mehmet ascended the throne, he was ruling the same Romans, Greeks, Slavs, etc. that had been ruled by his predecessor, Constantine XI.
Fourth and finally, the Eastern Orthodox church recognized Mehmet as Caesar.