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I hate to be that guy, but this is wrong.
The solar system is mostly in one plane because it formed from a cloud of gas. The cloud of a gas has some total non zero rotation and as the cloud collapses interactions flatten the cloud into a disk, where all of the planets formed.
This same principle applies to galaxies.
It's a combination of both, I believe.
The initial conditions had a definite rotational bias. This is preserved in the current orbital plane and direction.
On top of that, anything massively off that plane is liable to hit or interact with the material in the plane, given enough time. It will be flung around, eventually either out of the system or into the plane.
Stuff orbiting relatively close to the plane will have a biased pull towards the "average" plane. This will slowly flatten the orbits out.
All these processes take a lot of time. The solar system, in general, has had enough time to settle. The ort cloud and other outer bodies are still quite chaotic. We see a lot more off plane than within the traditional solar system. They experience the latter effects far less, and so take longer to equalise. They still have a bias towards the initial spin however.
I was explaining why the orbits are similar, not why orbits exist. You're arguing a different topic.
The original question was why solar systems and galaxies are in planes, and your explanation is wrong.
What do you even mean by similar orbits? Most orbits are circular for a totally different reason, and that is tidal interactions.
It's been a long time since I got my astronomy degree, but your version is what I recall also. Whatever small rotational perturbation in the initial gas becomes more pronounced as it coalesces in on itself and defines the plane of the star system. Planets form within this plane after it is defined, and they all travel in the same direction around the star.
Regarding galaxies, the most common spiral ones like our own Milky Way follow the same principle at a larger scale. But there are also elliptical galaxies, not to mention irregular ones. In an elliptical galaxy, there is a more random movement of stars in a cloud around its core. So they look more 3D I guess, to go back to what the OP was asking about. I seem to recall the most accepted explanation for how these form is from the aftermath of a collision between 2 spirals? So presumably, when our galaxy collides with Andromeda in several billion years time, the resulting combined galaxy may emerge as an elliptical?
I bet there's probably some sort of energy improvement for the orbits to roughly line up the way they do, sort of how geese will fly in V formation.