The basic theory of spirals is basically that you get spiral structure when the orientation of elliptical orbits shifts with semi-major axis. This leads to what appear to be spiral arms that have higher densities. In these high density regions you get collisions of gas clouds which leads to star formation. The star forming regions produce lots of bright, blue stars, which then make the spiral arms very visible in optical wavelengths.

In this case of the Oort cloud the galactic tides would be what are responsible for inducing the change in orientation of the elliptical orbits as a function of semi-major axis.

It's amazing, yes, and at the same time, it makes perfect sense.

(Somewhat) similar mechanisms are at work whether you're pulling together stars into a galaxy, hydrogen gas into a solar system or water towards the drain of your bath tub - a pull towards the center, the centripetal force, slight variations producing "artifacts".

IIRC the galactic spiral is believed explicitly to be not due to gravitational attraction so much as shock wave/traffic jam dynamics (transmitted through gravitational force ofc) -- not sure if that's what you meant by clumping.

Do we know why ie Saturn rings are not spiral-like? Ie due to their age (some relatively recently broken down comets) or some other forces that keep them spread evenly? Or just gravity is too weak amongst them for those smaller pieces of rock

It will be really exciting if we confirm one, then.

The spiral structure here is a hypothesis within a hypothesis. Whatever objects comprise the Oort Cloud, they haven't been directly observed. Scientists have inferred its existence from a variety of comets that seem related and have very, very long orbital periods, such as 200 years, or 2,000 years. So these comets are observed once-in-a-lifetime, or once-in-a-civilization, and the hypotheses say that they're being dislodged somehow from a "cloud of planetesimals" where a bunch more of them are found.

But this supposed cloud would be extremely sparse: plenty of space in-between the very small icy bodies, and individually, they're so much smaller, and so distant from the Sun, that they don't reflect enough light to our telescopes. They really don't send signals in other wavelengths, either, like a pulsar or quasar or something with an active powerplant.

This is beyond the Kuiper Belt, even; the Kuiper Belt, if it indeed be a belt, has offered us a couple of directly-observed objects, including Pluto and Charon.

So it's nice to conjecture and invent proposals for some kind of structure there, but the very existence and extent of the Oort Cloud is something that's been extrapolated and inferred from secondary evidence.