Can reproduce.

TL;DR Metaballs have a fundamental issue you can work around by creating a collection that contains a Metaball object with everything in it removed using Edit Mode.
You can link this collection to several scenes, and metaballs will work across them. Pretty hacky, but it does the job.

Here's my best assumption of what's happening:
The first metaball you add to any scene becomes the "metaball space origin" where Metaball geometry is generated.
Transforming this first metaball (moving it, rotating it, scaling it) changes the way geometry is generated for EVERY metaball object.
In fact, that first metaball IS where the scene's entire metaball geometry is generated. (Notice how selecting the geometry selects the first metaball!)

This is a problem, because Blender supports multiple scene where a metaball can be added.
Since the first metaball stores the geometry of all combined metaballs, if it is missing in one scene, you won't see any metaball geometry in that scene where it's missing.
(Which is exactly what this bug report is about.)

I propose that there be one "metaball space" per scene where geometry is generated. The metaball resolution settings would be in the Scene Properties menu.

https://docs.blender.org/manual/en/dev/modeling/metas/editing.html#meta-ball-object-families

When working with multiple scenes, take care naming your meta objects so the base is always in the same scene as other metas.
Failing to do so will give confusing behaviors (like invisible meta objects).

@Yevgeny Makarov (jenkm) If there are naming restrictions, shouldn't blender itself name them properly when a scene is duplicated?