The primary benefit is not the sideways thing.
I could imagine that the sideways thing could be the main benefit.
Imagine a ten story building with two lift shafts and 10 cars. In a quiescent state there's one car parked on each floor.
During light times, someone gets into a lift, the cars above or below move out of the way and then it travels to the required floor. The cars it has passed simultaneously shuffle up or down one floor and you're back to the quiescent state. In busy times there can be one up and one down shaft - but there doesn't need to be the same wait time, indeed the lifts could be positioned every other floor.
Biggest issue with this I can see is the loading/unloading time. Ideally you'd want to do something similar to the way some London underground stations work where people leaving the lift go out through a different door to those entering the lift. Which may defeat any space savings from having fewer lift shafts.
What I'm not sure about is whether the lift can safely move sideways while people are in it. Feels to me as though trying to stand in such a lift would be like trying to stand on a train or boat where the sideways forces require significant strength and coordination to remain standing unsupported.
If the cars have multiple doors then you could have four lift shafts in a square array and every car would have four available directions to move.
Of course, what you could do is open the doors and then rapidly move the lift sideways, ejecting the people out of the door (a-la pulling the table cloth off the table without disturbing what is on the table) and readying the lift for the next load ;-)