Well, I can draw a paralell between the star trek one.
In the game "Freelancer", everything happens inside a single "sector" of space. The sirius sector. (I imagine it has something to do with the starsystem(s) of the same name..)
Even though there are four "countries", so to speak. There's also a lot of independent or empty star systems within the sector.
Sector itself just means 'area'.
But I think it's more of a specific cluster of stars, when it comes to sci-fi/space stuff like this.
Which can be any size.
Just like, say, a valley. A country.
A mountain range.
You get the idea.
It's like how an industry can also be split into sectors. I think it's mostly a thing for convenience's sake.
In the case of this game, though, perhaps when mankind was still at it's prime, they designated various sectors of things that were similar or maybe political density or some such. Could also just be like how they split the US into states. "Let's put a box here and a box there of similar sizes"
The possibilities are endless
edit: I generally imagine it to be a handful of stars. Tops 24-48. (dozens?)
But what do I know? xD