It does! The solar system moves through the galaxy face-first: http://rhysy.net/Art/Illustrations/Solar-System-non-Vortex/

Think about a chef taking a ball of pizza dough and spinning it above their head.

It doesn’t turn into a big, hollow sphere, it turns into a round, flat pizza.

Same principle, except the ball of dough is a ball of stardust in space.

if you mean that planets revolve around the sun roughly in the same plane, as the star itself rotates, then look in to the formation of the sun and planets from the same disk of dust.. movement is tightly coupled to the formation itself.

In space there is no up and down.

It all formed from the same accretion disk. Very flat, all moving together dust and gas became the whole solar system

Someone fact-check me, but I'm pretty sure that there is two reasons for it. The first reason as to why all the planets are spinning in the same direction is because when the planets formed there were different planets, or basically small rocks that were going in different directions, but all the different direction ones hit each other and eventually one direction that had more planets became the only direction.

as to why the direction is not up and down, it doesn't really make sense in space as there is no up and down but there is the direction that the Sun is moving in throughout the galaxy and if that direction is up then the planets are going on the sides and not up and down and that makes sense because the Sun is pulling all of the planets in that direction, so they are dragging along, so they can't spin in front of it and then behind it.

EDIT: The reason all of the planets are spinning in the same direction is not what I wrote. It's actually about the fact that they all formed from a single gas disk. So they all formed from a disk that was spinning in a single direction. The collisions actually contributed to the growth of the planets and less so to their direction.

Here's a Minute Physics video that addresses the question: https://youtu.be/tmNXKqeUtJM

There is no “up” or “down” in space

I'm no expert, but it sounds like we just discovered that the galaxies are disc shaped because of dark matter pushing down on both sides. Not sure if that applies to solar systems as well though.

source: https://youtu.be/XNbqyUvOVLc

it does, we just usually look at it sideways

There is a gif I've seen that demonstrates this using a water balloon and a drill. The knot of the balloon is in the drill chuck, and when you turn on the drill the balloon spins into a (relatively) flat disk.

I'll try to find it and add a link here. Prepare for disappointment.

Edit: best I can do is an Instagram reel (from a great creator who covers lots of physics stuff) https://www.instagram.com/reel/DKLwz5TxmIu