The stars all orbit around the center of mass of their galaxy like asteroids orbiting the Sun in our solar system. If two stars had exactly the same orbit they would bein in the same place and be a binary star.
The orbit of each star system must be different from the orbits of its neighbors. Thus as they orbit the galactic center they will move farther and father apart and won't start to move closer to each other again until they have completed more than half of a galactic orbit a hundred million or so years later. But passing close to other stars will change their orbits, and they will pass close to a lot of stars during a single galactic orbit. Thus a group of stars which are currently close to each other will probably never get that close together after a hundred million years.
Here is a link to a list of the stars which have passed or will pass within 5 light years of the Sun within a few million years in the past and future.
This indicates the rate at which stars may move relative to each other.
So if two interstellar realms have a nice flat border between them, it is quite possible that after a few tens of thousands of Earth years, the border will be very bumpy as stars from Realm A move in between stars of Realm B, and vice versa.
This may be a problem for interstellar realms lucky enough to last for tens of thousands of years.
I don't know the time frame that will be covered by your single short story, or long series of epic multigenerational novels, set in that fictional galaxy. Thus I don't know whether the time covered will be long enough for that to happen, nor do I know how the space realms would deal with the changing borders when and if happens.
Added January 2, 2023.
Another answer suggested claiming a globular star cluster, whose stars are gravitationally bound to each other and travel together.
Here is a variation on that idea.
One possibility would be to claim all the stars in a moving group.
A moving group, in astronomy, is a group of stars that share a common motion through space as well as a common origin.
The stars in a open cluster would all be young, since it would disperse rapidly by astronomical standards.
Open clusters generally survive for a few hundred million years, with the most massive ones surviving for a few billion years.
So some advanced society could colonize an open star cluster, terraforming young planets to make them habitable. And as a star cluster gradually disperses, it will form a moving group of stars. The farther apart the stars get, the more ordinary stars will be between the members of the moving group.
I believe that the Sun is within the volumes occupied by the members of at least one nearby moving group - the stars in the moving group are in every direction as seen from the Sun.
As the moving group spreads out, there will be tens, hundreds, thousands, & eventually millions of more or less background stars within its volume for each star that is actually part of the group.
So a space realm could claim all the space within a distance of X light years or parsecs or whatever distance unit is used around each of its colonized stars, and claim as a border a number of flat planes connecting the spheres around the outermost stars it claims. Thus it would claim all of the stars which pass into the volume occupied by its moving group, and colonize their planets.
And that would be fine for the interstellar realm so long as the nearest other interstellar realms were hundreds or thousands of light years away with unclaimed space and stars between them.
The closer various interstellar realms get to each other, the more likely it is that stars will pass from the volume of one realm into the volume of another realm. Especially when the space realms directly border each other.
But as I said above, the distances between stars are so great that the time for a star and its planets to pass out of the volume of space realm to that of another space realm should be extremely long by human standards.
Thus it quite possible to write a long, long, epic space opera series which covers thousands of years, in which stars passing out of one space realm into another doesn't happen because the time scale is too short.