Planets are built from the material left over around stars
When a star forms, it doesn't use up all the material around it. What's left settles into a flat, rotating disk of gas and dust. Planet-formation models follow planets as they grow and migrate inside this evolving disk, and that disk is the raw material for planets [1]. Because the disk is tied to the star, anything that forms in it is naturally in orbit around that star.
This is why the pairing is so common: the star and its planets come from the same cloud of material. The planet doesn't get captured by the star later; it is assembled in the star's own disk. The models that generate synthetic planet populations treat the disk's mass and lifetime as the key inputs, and those inputs are set by the central star [1].
The star's size shapes which planets it gets
The mass of the central star matters because it influences how massive the surrounding disk is. Population-synthesis models show that the disk mass scales with the star's mass, and this scaling has a direct, large effect on the resulting planets [1]. Around stars about half the Sun's mass, large planets are scarce, and a slightly steeper-than-linear scaling of disk mass with stellar mass can explain that scarcity [1].
The same scaling helps explain another pattern: short-period planets are rare around stars about twice the Sun's mass, but only if those heavier stars also have shorter-lived disks [1]. In other words, the star doesn't just host planets; its mass and the lifetime of its disk help decide how big the planets are and how close they sit.
A planet can stay in orbit even as its star changes
The bond between a planet and its star can survive dramatic changes. A giant planet has been found orbiting a star that has already passed through its red-giant phase, showing that planets can persist through the star's expansion [2]. The star swells and changes, but the planet's orbit remains tied to it.
This reinforces the picture that the planet-star link is gravitational and durable, not a temporary coincidence of formation. Even when the star's outer layers change, the planet keeps orbiting the same central mass [2].
Why don't we see planets orbiting other kinds of objects?
The supplied evidence focuses on planets around stars and does not directly test planets around non-stellar objects, so the question of whether planets could orbit, say, free-floating or non-stellar bodies is not settled here [1][2]. What the evidence does show is that the formation pathway we can model—growth in a disk around a star—naturally produces star-orbiting planets [1].
There is also a detection angle. Planets are found by watching their host stars for tiny wobbles or dips in brightness, and multiple-star systems complicate this picture because planets can orbit different stars within a binary or multiple system [4]. Even in those crowded settings, the planets are still tied to stars. Studies of such systems find that planetary and stellar orbital planes tend to be aligned more than random chance would predict, which is consistent with planets forming in disks sculpted by the star system [3].
Sources used in this answer
Extrasolar planet population synthesis III. Formation of planets around stars of different masses
Population-synthesis models show that the scaling of protoplanetary disk mass with central-star mass strongly shapes the resulting planet population, helping explain the paucity of large planets around 0.5-solar-mass stars and the lack of short-period planets around 2-solar-mass stars.
A giant planet orbiting the 'extreme horizontal branch'star V 391 Pegasi
A giant planet has been found orbiting a post-red-giant star, indicating that planets can survive stellar expansion.
Orbital architectures of multiple-star systems that host transiting planets
In planet-hosting binary and triple star systems, stellar and planetary orbital planes show more mutual alignment than random isotropic orbits would produce.
ALMOST ALL OF KEPLER'S MULTIPLE-PLANET CANDIDATES ARE PLANETS
Statistical validation of Kepler's multiple-planet candidates confirms that almost all are planets, including cases where planets orbit different stars within binary or multiple systems.
Five planets orbiting 55 Cancri
Radial-velocity measurements of the nearby star 55 Cancri provide strong evidence for five orbiting planets on Keplerian elliptical orbits.