I textured an object in Blender because it wouldn't texture properly in Unity, and then imported the object and texture to Unity.
I don't know how to fix this, I'll put both pictures here.
Okay, so based on your screenshots..
You're going to select everything then add a modifier called "Solidify", then set this to something very small like .03. (Unity doesn't like objects that are just planes).
double check all your normals are facing out. Let me know if you don't know how to do this...
Go into edit mode select all edges, then right click, select "mark as seam".
Open UV editor window (should be split screen with Edit mode on one side, UV view on the other). In the Edit side select all, then go to UV dropdown menu and click "unwrap". You should then see your object unfolded into flat planes over on the UV window side. There's different unwrap options, like smart UV unwrap, etc. I think just "unwrap" has worked for me, but play around and there may be something that shows your object shapes in a less distorted way..
At this point, since your pattern is basically repeating. If you export the OBJ file and take it into unity, and you add the image file (make sure it's dimensions are perfectly square) it should receive the image file as an Albedo texture much better than in your screenshots. You can play with the 'tiling' and "X/Y offset" till it looks right (you might face issues with rotation though.
BUT