Within our galaxy, the general expansion of the universe plays no role, it is essentially something that happens to the whole galaxy and to its individual parts, like water to a fish, is invisible. It isn't until we start looking at other galaxies that we notice it.

Motions of stars in the galaxy are dominated by the general galactic rotatation. Think of it as a group of ants on a phonograph record. As the record spins, all the ants are carried with it, but they also are free to walk their own random motions on the record.

The average motion of all stars in the solar neighborhood is defined by the term Local Standard of Rest, or LSR. It is an imaginary point in orbit around the galaxy, and all the nearby stars, including the sun, also have random motions relative to the LSR. Think of the LSR as a dot painted on the phonograph record. But all stellar motions, proper motion across our line-of-sight, or radial velocities in our line-of-sight, are measured relative to the earth (the sun, actually). In other words, stellar motions measured from earth are corrected for earth's orbital motion, which reduces them to the sun as the origin. Those interested in galactic dynamics further reduce them to the LSR. If there is sibling to the sun near enough for us to see, it would have the same motion relative to the LSR as Sol does, so it would have zero proper motion and radial velocity. Otherwise, it would have drifted away a long time ago. Unless it is gravitationally bound to the sun, the random perturbations of 20-odd galactic rotations should have stripped it away a long time ago.

We do know of some relict clusters than can still be recognized by their common proper motions. The stars of the big dipper have a common origin, and I recall there is a "stream" of stars scattered across the winter sky which have a common birth cluster. Many of these "stellar associations" have been discovered, they are the last remnants of open clusters that have almost dispersed. But they are short-lived objects, the shadows of the original clusters. Another galactic rotation or two and they'll be sheared apart. The sun is too old to be likely to be near its siblings.

The most dominant fact about all this is that the galaxy is 100k ly across and 1k ly thick. At visual wavelengths, we can only see a couple of thousand light years into the disk. Radio, and to some extent, IR, can penetrate the galactic dust haze and allow us to see further, but as a rule, we can only study in detail a very tiny portion of the galaxy. The sun is near the centerline of the disk, about 30000ly from the nucleus. So we can see above and below us the 500 or so ly to intergalactic space and beyond. But looking into the Milky Way, we can only see clearly for about 2000 ly.

We are like 1 pilgrim at Mecca, vaguely aware of the vast crowd around us, but only the faces of our immediate companions are recognizeable.