Airline pilot here: passing another aircraft opposite direction on the same lateral track with only 1000ft of vertical separation is about as normal as it gets. I’d say that it has occurred on almost all of the flights in my career so far. In certain areas of poor ATC coverage we sometimes use SLOP (strategic lateral offset procedures) to add a bit more margin for a height error.

Edit: fun fact, you can actually hear the passing jet from the flight deck just after it has passed, a slight whoosh can be heard. Depending on conditions you can also feel a brief bump when crossing their wake (since the wake descends).