The gravitational region around the planets isn't hard to calculate, sometimes called the Sphere of Influence, sometimes called the Hill Sphere. They're calculated differently but they define pretty much the same idea.
The actual long term stable region is somewhere around 50% of the Hill Sphere. A gas giant like Jupiter simply doesn't have a large enough stable region that it has gravitational control of to have it's own Kuiper Belt or Oort Cloud equivalent. Also, proximity to the sun and the gravitational stability of the trojan points is a factor. What actually happens with a Sun-Jupiter system is that the stable regions are the trojan points and (because of Saturn) the Cis-Resonance orbits but not trans-resonance (Saturn is too big and too close to Jupiter, so it disrupts trans-resonance). Jupiter's crowded Trojan points and Hildas (mostly Cis-resonance) are a little bit like it's kuiper-belt equivalent and that's a result of the sun-jupiter system, the sun being dominant and jupiter, a thousandth the mass of the sun, but large enough to be dominant among the planets.
It's worth pointing out that at Jupiter distance from the sun, the solar system is quite a bit more crowded and has largely flattened out into a plane, unlike the Kuiper belt, so it's not a real comparison to the kuiper belt, but that's where orbiting objects tend to collect, cause those are the most stable regions. The Kuiper belt and Oort cloud aren't really regions as they encompass the entire sphere around the sun at certain distances.
It might be possible, I would think, for a gas giant in the right situation to have something like it's own Kuiper belt and Oort Cloud, but it would need to be hugely distant from it's star and far from other large planets and preferably quite massive. The limitations would still be no more than 1/2 of that planet's hill sphere, but it's theoretically possible, I would think, under the right circumstances. Even so, such a situation, the Trojan points and trans and cis orbitals might still be more common than any possible kuiper belt or oort cloud equivalent, so I'm just speculating that it might be possible. I'm not 100% sure.
If Jupiter or Saturn were considered stars their rings would be called 'asteroid belts', their spherical moons '(dwarf) planets' and the other satellites 'asteroids'. This is particularly evident with Jupiter's rings that are broad and not visible to the naked eye, so they're more like the asteroid belts around a star than the rings of the other three gas giants that are visually outstanding.
There is no natural satellite that orbits above or close to a pole of a gas giant, as far as I know.