>Why is it happening so? Sounds way to simple to be true in something so complicated as space. Even electrons have 3-dimensional orbits but planets somehow don’t?

Astronomer here:

I would like to clarify that, while each planetary system is on a plane, and the galaxy is on a plane, they are not all the same plane. The planes of the planetary systems are essentially random, and not aligned with the galaxy. Our own solar system is not aligned with the galaxy either.

The reason why spiral galaxies and planetary systems come out in planes has to do with the fact that they are all born from a gas cloud. Any initial gas cloud has some initial angular momentum. As it collapses by its own self-gravity, it has to spin faster to conserve angular momentum. This by itself is not enough to make the gas form a disk. The last ingredient is that gas in space actually behaves like gas ---- it feels pressure, it emits energy. So the initial gas cloud is a "blob" with a lot of random motions, pressure (think gas drag) dissipates most of the random motions, leaving only the "average" motion, which would be a rotation in some direction corresponding to the net initial angular momentum of the blob.

Stars in the galaxy are on a plane because stars from form the gas, and the gas was on a plane. Planets are on a plane, because planets form from the gas (well, Earth forms from the 1% of dust inside the gas) and the gas is on a plane.

gravity of all the planets pulling on each other pull them into a stable, flat plane. Imagine all the planets but one rogue one are in this same plane, like circles on the surface of your desk.. The rogue planet orbits at a big angle, so even if it is going in a circle, half of the time it is above the desk, and half the time below (like this: https://upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Orbit1.svg/1280px-Orbit1.svg.png - the grey is the plane all the planets are in, apart from the rogue one which is the yellow orbit) . When it is above the plane, all the other planets are located below it, and pull on it with gravity, down towards the plane. When it is below the plane, the planets in the plane are all above it, and they all pull it up. Over a very long period of time this brings the rogue planet closer and closer into the plane. Once within the plane, the planets still pull on each other, but there are no forces which would cause them to leave the plane again. Over time everything will settle into this single plane as its the only true long term stable solution.

I must admit I'm terrible at physics so I don't fully grasp it, but seems like it's because the solar system formed out of a spinning ball of gas and over time the spin caused it to flatten out (like a pizza dough, I guess?).

https://science.psu.edu/science-journal/winter-2021/FlatSolarSystems

As for why atoms aren't similarly flat, it seems to be because at that small scale the gravity is a negligible factor compared to the other forces, such as electromagnetism. The electrons repulse each other.

https://www.quora.com/Our-solar-system-spins-on-a-flat-plane-so-do-atoms-also-do-this

If you spin a ball of pizza dough it flattens into a plane.