It is mathematically possible for a universe to be shaped this way, but not our Universe. Our Universe is as close to flat as we can measure right now, though it’s only possible for it to be very slightly curved, considering the wiggle room we have remaining on our measurements.
The universe that you describe could be round, donut-shaped or cylindrical; some shape where at least in one direction, it connects back to itself. These aren’t your only options for a universe - you could also invent a saddle shaped or other, more exotic shape to place your universe in.
For now, let’s roll with a cylindrical universe. And let’s put a star somewhere on the surface. If the light from this star is going along the length of the cylinder, all it can ever do is go out, because the surface is flat in that direction; there’s no curve or loop. This flat, uncurved behavior is how we believe our Universe behaves in every direction. Light in our Universe departs its star, and travels in a straight line forever (as far as we can tell) unless it is intercepted by another astrophysical object, another star, planet, or telescope detector.
However, the light that leaves our star in the cylindrical universe has one other option. The light that goes in the other direction - around the curve of the cylinder - will also travel in a straight path. But this path loops back on itself, and if the light doesn’t hit anything else, after it has completed its tour of the cylinder’s circumference, it will arrive back where it began, on the other side of the star, delayed by the length of time it took to do its loop.
What happens if you make your universe spherical? It’s a very similar thing, except now every path that light can take will loop back onto itself, given enough time. There’s another curious thing about the light this time, though, which is that the beams of light, even though they’re all travelling “out”, will all cross each other at some other point on the sphere. If the star was on a flat surface, these beams of light would only ever get further apart; there’s nothing that would ever curve the light back towards each other.
In our Universe, we know that there’s no bending of the light as it comes through space (this is from an analysis of the map of the oldest light in the Universe) beyond what you would expect from gravitational forces. This lack of a large scale bending rules out the spherical and saddle-shaped options, and all that’s left are the ones which can be considered flat. While we can’t observe the entire universe to objectively figure out what the global shape of the entire thing is, we know that on the scales of the observable universe, our Universe is pretty darn flat.
How do we know that the Universe isn’t a tightly rolled cylinder? Well, we can’t rule out a gigantic cylinder, but it would have to be so large that we couldn’t ever detect a difference between light going “out” along the length of the cylinder and the light going “around”, because as far as we can observe, the Universe is the same in every direction. If there were a preferred direction, where the Universe appeared considerably younger than in the other direction, then we’d get suspicious of a cylindrical shape. But since there’s no evidence for that, we usually describe our Universe as an unwarped, three dimensional, grid. And with that kind of shape, we don’t expect any of the light from the distant universe to be taking a looping path to show us our own Milky Way.
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