I didn't really want to answer this one because there may be some new information that I'm not aware of and, to my understanding, the shape, mass and content of the Oort cloud is a subject of some ongoing study, so I invite correction.

The formation of a solar-system is pretty complicated and there may still be significant unknowns on the process and with the Oort cloud, uncertainty on the precise shape, mass, density distribution and origin, whether it formed with the solar-system or whether much of it is captured. I wanted to begin with the unknowns.

We can't see the Oort cloud so we can't measure it directly. Estimates can be made by observing very long period comets that fly into the inner solar system and by extrapolating their orbital periods, an estimate of the contents of the Oort cloud can be made. One problem is, that estimate is based on very eccentric orbits only, because those are the only ones we see that enter our telescope's range.

This is one of my favorite minute physics videos for it's simplicity and the fun diagrams they use. The gist of the simple answer is that a cloud or nebulous mass of debris and dust, has a fixed angular momentum and a fixed orbital plane, so as the cloud of debris gets a push (usually from a not too far away supernova), and the debris begins to condense into a proto-solar-system, it maintains the orbital plane and angular momentum and as it condenses, it rotates very rapidly.

The rapid rotation isn't all that relevant to your question, but that's why not all the matter can fall into the star, because there's usually too much angular momentum. The same happens with gas giant planets, which is why Jupiter, for example, has 4 large formation moons (where as the Earth - somewhat smaller region where it formed, no formation moons, but it has an impact moon). I'm making this more wordy than it should be, but the point is, stuff orbits the sun because of angular momentum. The sum of this angular momentum has an orbital plane, and each individual object has an inclination to that orbital plane. When planets absorb the matter in their orbital regions, the up and down directions, or inclinations of the objects in orbit tends to cancel out, but the net angular momentum and orbital plane remain constant. (This is mostly, but not entirely true - for example, Planet 9, if it's eventually discovered, might explain why the 8 inner planets, on average don't line up with the sun's rotational plane. Planet 9 may have taken some of the inclination in one direction with it as it was thrown far outside the solar-system leaving the inner 8 planets with an inclination in the other direction relative to the sun. When (if) planet 9 is discovered, then we can check if it balances out our solar-systems tilted inclination.

But it's the collisions that help the planets line up along the orbital plane of the solar system, because the ups and downs mostly cancel out. If there are no collisions, then there's no cancelling out of the ups and downs and the objects in orbit remain in a kind of nebulous blob assortment - which over time, is probably best represented as a sphere.

That's not the entire answer, however. Take the asteroid belt for example. There was probably not sufficient collisions to make the asteroid belt flat, and Ceres is probably not a failed planet, because it probably came later. Based on Ceres' density it probably came from further out in the solar-system. It may have started out as a moon (perhaps kicked out of Neptune's orbit by the miss-behaving Triton) or a dwarf-planet originally in the kuiperbelt. It's density is too low to have formed by collisions in the asteroid belt.

So the Asteroid belt is flat (er mostly) likely due to gravitational shepherding from Jupiter and perhaps, by a very strong magnetic field from our young sun,

A proper answer of our estimated shape of the oort cloud would be based on a survey of all the long-period (or highly eccentric) comets that we've observed and I'm not that interested in doing the research on that, but I would guess that there's enough variety of orbital inclination to support the generally circular shape - because I don't think the circular shape would be used so often if it didn't reflect the observation of long period comets.

The Kuiper belt, for example, (and I couldn't find a really specific answer to this), but it appears to be somewhat flat (Pluto obviously has a pretty high inclination). Rather than say flat or spherical, the proper term is inclination distribution. No distribution = flat. Full or high distribution = sphere.

The Kuiper belt's relative/somewhat flatness may be driven by mostly Neptune sheparding rather than collisions (again I'm not precisely sure). In fact it was the regularity of some inclinations of the most eccentric kuiper belt objects (just passing through in other words) that lead to the planet-9 theory in the first place. If there's large planets around, they assist in the shepherding and flattening of smaller objects in their orbital viscinity.

In the case of the Galaxy, to my knowledge, there's insufficient collisions to explain the flatness of the Milky-way (it's basically pizza shaped, with maybe a small ping pong ball or large marble in the middle). It's my understanding that the Galaxy was flattened by a very strong magnetic field more than by collision (somebody correct me if I'm wrong).

That's the extent of my knowledge at least. I invite someone smarter than me to answer this as well.