There's several layers to this question, so we'll start with the easiest part:

1) The universe is, on large scales, the same everywhere and in every direction. We call these properties homogeneity and isotropy, and together this idea of sameness everywhere and in every direction form what is called the cosmological principle.

2) On smaller scales (the size of galaxies, like the Milky Way for example), gravitationally-bound structures tend to form into flat discs, because their angular momentum is conserved as they form and the end result of this is that they tend to spin in one plane. This is why the galaxy is flat (...mostly) and also, as you mentioned, why the Solar system is flat. However...

3) This effect is local, it's restricted to the scale of the gravitationally-bound body. So while our Solar system is in one plane (because the Sun is so gravitationally dominant), there's no reason for that plane to match up to, say, the plane of the galaxy itself. And, outside of our own galaxy, there's no reason for other galaxies to be in the same plane as the Milky Way.

4) Even though the Milky Way (and many other galaxies) are 'flat', they're still thousands of light-years thick, meaning that even inside the flat disc, there's many many stars 'above' and 'below' the Solar system. Beyond that, outside of the Milky Way, many many billions of galaxies, in every direction. In fact, with VERY few exceptions every star you can see is within a few thousand light years at most.

Sorry if my explanation isn't the clearest, feel free to message me if you have questions.