How exactly does this work?

looks like an autorigger (automatically create the joint chains and attach them to the model), animation retargeter (take existing animations from one joint chain topology and apply to another joint chain topology) and motion interpolation system (seamlessly blend between canned animations so going from a walk to a run does not look janky)

and they seem to be rolling things out per object class, currently they do wheeled vehicles and quadrupedal animals.

Yes

"How should this (thing) likely move? Apply movement parameters."

That's an oversimplification of what they are doing. If it was direct vertex manipulation for animating the mesh without any intervening rig structure (like how alembics work) your description would be be more on the money.

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Is there any project for 2D animations? I'd be happy with something doing just very basic animations of characters portraits for example like in Stellaris.

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I'm looking for something working with 2D rigs to use them as game modifications. I haven't seen such a model around, only stuff with 3D rigs.

an autorigger like mGear or Advanced Skeleton + an afternoon of weight painting using ngSkinTools will get you comparable rigs to what's been shown. The trickiest thing to rig is the face (either joint or blendshape based) and or get a full sim muscle setup going. Neither of these are shown.
For something low poly, stylized and simple like the fox that is shown it's no time at all with current tooling.