Objects must share same material
For some reason, even though i followed exactly the steps you showed, at some point i got stuck with only one object being painted.
The rest of them would not respond to texture painting, and it took me a while to find the solution with the help of google and blender communities.
I guess you talked about it at some point but i didn't want to go 3 videos back to find out where i missed something.
So i'll just leave that here :
Objects MUST share the same material if you want your global UV to work on all of them
I guess i just played around with the materials at some point and various objects had a material.001 002 etc 007.
The same logic will apply on Unity