Certainly merging two laser scans into a single point cloud is doable. Here's an example tool and you can easily concatenate said point clouds too.
However I'd like to suggest that you reconsider some of your assumptions in your question.
Each LIDAR pointcloud will definitely have unique sections in their FOV, but there will definitely also be overlap.
Do they really overlap? Aka they point to the same point exactly, or do they overlap only if you're assuming that they are planar. And also if they see the same object from different angles does that give you more information? Are they vertically alligned and planar? Do they have different viewing angles, different heights.
I think things like ICP and the like are out of the question since the interest is to have one pointcloud per time sample to use for the perception pipeline, the interest here is not for mapping.
ICP can be applied to any sequence of point clouds. There's no rule about when or how to update. Many implementations may be optimized for taking repeated scans from the same scanning laser sensor as that's the most common use case.
A fused cloud may be better for doing things like general obstacle avoidance.
Using each scan individually for sequential ICP may be the best way to track odometry. Some algorithms can take advantage of the semantic meaning of getting one consistent laser scan, while some can benefit from having a less structured fused data source. And others may actually want the multiple feeds fed in in the raw format and they will do their own fusion.
With that said I would suggest that both it's not an either/or question but what will serve your application best. You may want to do both feeding different forms of the data to different algorithms to enable them to operate optimally.
Is the standard practice just to add them on top of one another? What should one then do about overlap?
It completely depends on how you plan to use the data. There's a whole field of study called sensor fusion which I would recommend you do some reading on and ask more specific questions when if you have them. This is too large a scope to answer here.