While I don't disagree with the history of African hunting and it's effects on firearms development that others have posted above, there are mechanical design factors that also influenced things.
Both SXS and O/U design's use a standing breech, and a hinged barrel that is held in its closed position during firing by some form of locking bolt.
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When the cartridge is fired, it pushes back against the breech face and forces the barrel forward around this hinge point. In this picture of the Browning Citori O/U, you can see how much farther the top barrel is from the hinge point and locking bolt (circled in yellow) than the bottom barrel.
The closer the barrel centerline is the the centerline of the hinge, the less the torque force applied to the locking bolt, but the farther the barrel center is from the centerline of the hinge the more torque force is applied to the locking bolt. This is why most O/U shooters will only load and fire the bottom barrel when shooting singles Trap, and will only load the top barrel along with the bottom barrel when shooting doubles. It saves wear and tear on the locking bolt and locking bolt recess.
Just as a reminder, shotguns typically use less than half the pressure of most rifle rounds, and place a much lower overall force against the breech face.
The SXS, having it's barrels set very low and close to the hinge centerline places less stress on the locking system than an O/U.
However, the SXS, having it's barrels offset to the left and right of the hinge centerline presents a completely different set of problems. When the right barrel is fired, it's torque forces push the breech end of the barrel to the right, and the muzzle to the left. When the left barrel is fired, it's torque forces push the breech end to the left, and the muzzle to the right.
This is why most older SXS shotguns that have thousands of rounds through them will be very loose and can be twisted and rattled because the hinge pin, locking bolt and it's recess have become worn.
Most makers of quality Double rifles use 2 underlug type locking bolts, and an additional top locking bolt like a dolls head or a Rigby rising bite.
Making a strong double rifle that will stay tight for thousands of rounds is no easy task. This article gives a good description of some of the locking systems.
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Because of the torque forces involved, and the difficulty of loading an O/U bottom barrel under stress vs a SXS, I think the Side by Side makes more sense for the intended purpose.