Is it possible to save shaders and node groups to the library? Like the ones found here:
Yes, you need to save a blend file with the materials marked as assets into your asset folder. From that file you can move the materials into the category you want (save the file again after moving).
You can not just save a material, geometry node modifier, or any “node group” on its own into the asset folder, you need to save a blend file that contains that material, geometry node modifier, or “node group” and mark them as assets.
Asset libraries are basically folders that contain blend files with assets, you can make your own library folders wherever you want and add them to your asset browser in preferences. (“Asset Libraries” section)
Asset libraries can contain many different types of assets you can move different assets into different categories ( materials, lights, Geometry node groups etc ) you can also make your own categories.
The whole asset system can seem complex at first, I recommend you watch a tutorial for the asset browser and libraries.
I thought that BlendKit had an option to do this for you automatically.
Not sure I do not use blend-kit so i do not know, I meant assets in general, I make my own.