How do they manage to move multiple “cutouts” independently at the same time, using the mouse? Especially when they’re overlapping. Green screen multiple recordings, and then merge them? Or is there a simpler explanation?

Yes since the background image is static you don't need a greenscreen. From watching just once, I think the guy never overlaps his clone(s). So you make two separate versions where their bounding boxes won't overlap (requires some planning) and then just show the left two-thirds of the bg image with guy 1, and the right one-third with guy 2, divided by a simple vertical cut.

And the overall zooming/panning is done after the merging. That's how I'd approach it.

The video is highly edited, it just appears to be a single screen recording.

That was… hypnotic, no other word for it. Thanks for the discovery.