How to make edges of adjacent objects be visible?

On the picture below I have planes inside of the other planes, but they’re became invisible after material were applied.

Did you try?
Object > Viewport Display > All Edges

It seems you misunderstood me, I should see edges while object stays colorful and solid. It’s something to do with materials.

Perhaps you mean the following (see picture).
X-ray view is only visible in solid mode, where you can’t see the material, or in edit mode, where the material is visible too.

please show other screenshots of your object, maybe pack the texture and share your file…

I need edges be visible not only in Blender, but everywhere after export. It’s not about view settings in Blender specifically, it’s about making something with material - how you gonna see it if you open it in any other application.

There are different methods to realize it.
For example, you can use the UV as displacement map:

  1. Export UV layout
  2. Open in GIMP (or similiar)
  3. Select by color
  4. Selection grow (2 to 5 pixel)
  5. Fill with black
  6. Invert color (black becomes white)
  7. Fill background with black
  8. Remove alpha channel
  9. Use image in Blender as displacement map

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