"After the conquest of Constantinople, Mehmed II crowned himself the Caesar of the Roman Empire."
Well yes, that's how feudal systems work. If you take over a country, you get to have its titles. This is how e.g. an English king could conquer Scotland and crown himself king of Scotland. It doesn't mean he always identified as Scottish.
The language "reclaim the Roman empire" is really a stretch since it implies that the Ottomans had some sort of self-image as the natural heirs to Rome before they took Constantinople. Is there any evidence for this?
Why would they need to conquer Constantinople to do this, given that the Byzantines weren't trying to conquer, y'know, actual Rome?
I'm not saying you're wrong. People come up with lots of rationales for wars. And claiming historical right is a popular one, but that goes alongside other motivations as well.
My outside-view assumption would be that this is modern-day politically-motivated interpretation. These days, progressives rules the universities and the history departments. Therefore, they want to advance progressive goals. The see Muslims as a victim class, which means they don't want Islam to be seen in a negative light. Therefore, they would want to avoid historical interpretations wherein Muslims are conquering for self-understood religious reasons like ISIS did so recently. So I think they'd come up with something like, "Oh no, the Ottomans just wanted to reclaim their Rome."
Any time you read history you need to take into account the incentives faced by the people writing it.
In any case, when I refer to Islamic invasions I'm referring to the whole string of them going back to the Abbasid Caliphate - the conquest of Egypt, Arabia, Persia, all of north Africa, Spain, India, etc etc motivated by a self-declared doctrine of holy war (itself used for political ends by individual rulers, of course). I see Constantinople 1453 as just one small episode in this 1400-year story.