Shouldn’t we only be able to see other objects by looking out from the “sides” of our planet in line with the orientation of the flat universe?
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Flat and 2D are different. If the universe was 2D then yeah we could only look out in a circle instead of a sphere. When people say the universe is flat they mean it’s not curved. For example if the universe was curved like a sphere if you went far enough in one direction you would end up back where you started. That doesn’t happen so we say the universe is flat.
Math and physics people like to make up their own meanings for words and then complain that normal people don’t understand STEM subjects.
When they say “flat”, they mean “not curved”. And by “curved”, they’re talking about a kind of 3D curvature that’s difficult to picture.
Like, ever see a picture or video of a simulated black hole, perhaps in a film like *Interstellar* or in a video game? Notice how images of stuff behind the black hole get all funhouse mirror distorted? Like that. That’s 3D space curvature. When astronomers say the universe is flat, they mean the universe on a broad scale *doesn’t* look all funhouse warped like the black hole. As far as they can tell, it appears to just be the plain, boring, infinite(?) straight 3D grid line shape you’d intuitively expect it to be.
In this context flat means not curved. A parallel line is always parallel. Compare to a globe shape, for example, where the longitudes are not equal between latitude