Grelle There are situations when you need big and small versions of spine object in one project. So, it when I need small object, I dont' need big texture, scaled to small size - I need different texture with smaller atlas size.
It is obvious, but question is: how supposed to work with spine objects?

It depends on what problem you want to solve by using different atlas textures. If you just don't like the look of your high-resolution textures when viewed at small scale and e.g. want to avoid the blurry look of generated mip-maps, you could either change the mip-map bias at your textures (via the TextureAtlasAsset), or even specify your own mip-maps and still use a single texture as described in this forum thread. If you didn't have mip maps enabled and dislike the pixelated look, please have a try enabling mip maps at your atlas textures.

If you want to avoid loading higher resolution textures until visible at a certain scale, you could use mip map streaming (see the documentation here).

If you want to avoid downloading the high-resolution textures in the first place, you would need to use Unity's delayed-loading mechanisms such as Addressables, AssetBundles, and the like, which is a different topic (you should find some discussion regarding delayed loading here on the forum).

Grelle But this method is unsafe because of human factor. When you have 3-4 different sizes, you must not to forget export all 4 folders when change some animations, and not to mistake with folders and so on.

In general we highly recommend automating exporting your skeletons and not exporting them manually via the Spine Editor UI each time. This also helps when you suddenly need to re-export all your assets after a Spine version upgrade. You can find more info in the following forum posting:
https://esotericsoftware.com/forum/d/12408-please-tell-me-required-re-export-wont-happen-again/6

Grelle So if I need some prefabs for disfferent places, what I supposed to do? Create SkeletonRenderer (whatever I need) and use SkeletonRendererCustomMaterials for overriding in every place where I use my object?

Using SkeletonRendererCustomMaterials is a valid approach to replace a set of textures with their lower-resolution variant.

Grelle May be I don't understand how it works, but it is looks heavy for performance - override material at every update, so I wondering: is it possible somehow "cache" , what texture is used for gameobject? Or it is not heavy at all?

Replacing textures is not heavy at all performance-wise. However, if you want to avoid loading the high-resolution textures in the first place, you would not avoid that since everything referenced indirectly by the SkeletonDataAsset will be loaded when the SkeletonDataAsset is loaded by your scene (normal Unity rules regarding loading apply here). You could use the low-resolution AtlasAsset variant at your SkeletonDataAsset and then programmatically replace the low-res textures with the higher-resolution texture variants when needed.

Grelle Glad to hear it is helpful.