Ok, so no analogy is going to be perfect, but this gets the point across very well. Imagine our universe was 2D. Everything we know is living on the surface of a sheet of paper. You drip a drop of ink into the middle and BANG! A circle starts expanding outward in all directions equally. This creates a curved shape: a circle. A circle is curved. But the piece of paper is still flat. You can pick up the piece of paper, giving it curvature, rolling it, crumpling it up, whatever, but the circular splot on the 2D surface of the paper hasn’t really changed.
Spherical vs flat space refers to the curvature of spacetime itself. But any object (including our universe) which lives in the 3D space can be any shape (including spherical) without changing the overall curvature of space.
The universe ≠ space. I think that might be the misunderstanding. The universe is a thing that exists in space. Just like Earth is a thing that exists in space. The Earth is spherical (ish) but it doesn’t mean that space itself is spherical.
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