When the institutions of the catholic Church were created it was not onto a blank space. While in the first two centuries Christianity was, to an extent, a subversive an occaisionally revolutionary movement, when it was legalized it was bent into the shape of the Roman empire and became a true imperial institution. The most obvious example of this is the geographic arrangement of Church territories into dioces, which matched the imperial divisions instituted by Diocletian. Although there is certainly good, early precedent for organizational divisions of the Church, the specific way in which it was done was very distinctly imperial.

That being said, the idea that the Church is actually a continuation of the empire rather than an institutional fossil of it is not terribly justifiable (if nothing else, the Orthodox Church has at least as much claim to legitimate continuity). I would hazard a guess that someone saying this is perhaps attempting to grant historical legitimacy to the Church and overstating it a few touches.

Related question: was the spurious Donation of Constantine ever widely believed, and if so what impact did it have?

St. Augustine of Hippo summed it up quite nicely in "City of God" written in 426 CE.

Rome itself had been sacked in 410, and confidence in the old imperial institutions was flagging. What was a person to place their faith in when their eternal empire was teetering on the brink of oblivion?

At the center of this tome is the idea of conflict in perpetuity between the devil and God here on Earth. Each side has their forces, military, political and social, that engage with the other side in a never ending struggle for dominance. The Roman Empire was the vessel that birthed Roman Catholicism, a good thing that brought everyone closer to God and the forces of good. Now that the empire was about to come unglued, the church would step into the vacuum and be the new force that binds society together so that God may one day triumph over the forces of the devil.

If we go by this logic, Republic of Turkey has much better claim as modern day successor of Roman Empire because Byzantine Empire was direct successor of Roman Empire (Eastern Roman Empire atleast) then Byzantine Empire was usurpred by Ottoman Empire which adopted its capital and acted as it's successor and ultimately Republic of Turkey is modern day successor of Ottoman Empire. Even the idea is absurd.