That was a long tried and error run, but I managed to solve this.
The problem must have been with some sort of non removable history of a copy model of blend-shape.

So to not create all of my blend-shape models from scratch I did the following:

1. I deleted history and non-deformer history of all blend shapes and original model;
2. I created 20 copies of original model (because I will create more blend shapes in future, and I do not want to copy original model from the point where it was modified by other blend shapes... The reason I mentioned at the beginning);
3. Delete history again;
4. Then I selected one of older blend-shapes and copied attributes (transfer attributes) to the new selected copy of original model (New blend shape copy) to copy vortex points;
   (Options in picture)
5. Then I deleted history of all of copy and original models;
6. Finally I did "blend-shapes" from copies;

All works inside fbx file and Unity game engine that I am exporting those to.
Hope it helps, if not write to me, we will figure it out!


Apparently much simpler solution is to just delete all blend shape models (by delete button, not some fancy way) just before exporting to FBX. Then when it is exported undo if you still need them.