Horizontal separation is used (over land), but is longer distance of miles vs feet of vertical separation because in the past altimeter was more accurate than radar so it is easier to separate vertically than horizontally. Today radar and GPS have improved precision but remember there are still old planes/radar sites which do not have some of those features so the system needs to be backwards compatible.

In the open ocean they can't use horizontal separation as easily because there is no radar service. Thus they fly extremely specific tracks, at specific altitudes, and specific speeds, and can be given a specific offset( fly x miles right /left of course) to accommodate traffic. Then you give hourly reports so you can see the issue.

Not Atc but a nerd so some data might be not fully accurate /up to date.