It is the old claim of the Russian Orthodoc Church. With the fall of the Byzantine empire to the Turks, the Russian Orthodox rulers declared themselves a successor to Byzantium, thus claiming to be the successor to Rome (Rome- Byzantium- Moscow). (
https://en.wikipedia.org/wiki/Moscow,_third_Rome)
The Muscovite Romanov dynasty had been pushing this concept for centuries, and Putin also became addicted to their Kool-Aid. He imagines it to be his mission to restore the imperial grandure of Mother Russia.