I recently learned about the dominance of the Île-de-France over the rest of the country, and that into the 19th century less than a quarter of the population really spoke French.
But we never hear of the "Parisian Empire" or anything like that. My understanding of the term "empire" is that it purports to refer to civilizations with many disparate cultural groups under a singular central authority, like how the Romans ruled over the other Italians and Mediterranean peoples, or how the Ottomans ruled over the other Anatolians, Syrians, etc. Is this term used very arbitrarily or is this distinction mostly definite?
The meaning of the word "Empire" and its applicability in different times and places is a very frequent topic of discussion between historians of periods where empires feature, and a discussion I have often indulged in with other flairs here.
The basic things we tend to come up with are:
An empire has a core and periphery.
The core, typically, has a "core culture", which forms the basis for a related but distinct "imperial culture", which can come to influence and replace both the core culture and peripheral cultures (e.g., Roman Imperial culture becomes dominant over other Italic cultures and transforms the local Roman culture as well).
The territory outside the imperial core exceeds what would be considered a "normal kingdom" or equivalent administrative unit. The monarch generally makes some claim of possessing authority exceeding that of an ordinary prince, perhaps demanding that such princes swear fealty to him.
The last point, especially, tends to lead to inconsistencies and apparent paradoxes in how empires are defined. For example, when Anglo-Saxon kings like Alfred the Great began unifying the various Anglo-Saxon kingdoms, one could reasonably assert that they created "imperial" states. But the imperial culture of those states transformed over time into what we think of as "English" culture, and following the reorganization under the Norman conquests and whatnot, it no longer makes sense to speak of an Imperial state, as England is no longer recognizable as a set of kingdoms.
We see something similar in the ancient Near East. Sargon asserts some level of hegemony over multiple cities, and so he is often said to have founded the "first Empire". But after the rise of the Neo-Assyrians in the first milennium BC, which completely changes the dynamics of imperialism in the Near East, something like Sargon's state wouldn't have been very impressive at all.
So, it's all very context- and perspective-dependent. Generally, when we focus on an "empire" we want to focus on topics like cultural diffusion, administrative unity, commonalities in ideology, and so forth. Sometimes we also want to shift focus away from this, and look at the parts as self-contained units. So while you will most often read of the Persian or Achaemenid Empire, I think it's instructive to take a cue from the Persian kings themselves and speak of "the realms under Achaemenid Dominion" - they didn't see themselves as running a single "empire", but as possessing political and military power over a set of twenty-odd "lands".
I think that France is a little bit too out of my wheelhouse to go on at length about, but from what I know, I absolutely believe there may be contexts where speaking of a "Parisian Empire" could be helpful, just like it's sometimes more helpful to speak of e.g. the Burgundian Realms as a thing in themselves. Generally, late medieval and earliest modern France is a kind of textbook example of how messy "composite kingdoms" can get - it's not entirely different from the spaghetti sauce in a speckled-egg-bowl that is the Holy Roman Empire in this respect.