Scientists don't actually know what the shape is, this includes the visible portion. They just have assumptions of what it can be based on math. At least that's what I gather from Wikipedia when I went to verify my comments.In regards to higher dimensions, measurements show the universe is "flat." That means if you go off in one direction, you never loop back to where you started. We don't know for sure though since the universe could just be mostly flat. It's just an assumption based off the data, and I think this is what you're thinking of.
The visible portion is just a term describing the area of the universe that has impacted us up till that moment. Things past the observable universe were more than 13.8 billion light years away at the start of the universe and thus their light/casualty hasn't reached us yet. By definition the observable universe has to be a sphere. The universe is expanding so the visible portion is larger than 13.8 billion light years across but that doesn't affect the geometry.