Exporting 3D as .obj or .fbx, texture is not stored?

I am making 3D models for a friend who will import the models into Unreal Engine, I have read that Unreal supports .fbx or .obj and although he has successfully imported the 3D models, the paint which I apply to them does not transfer… I notice that the paint or “texture” that I use is saved in an external folder when I export the model.

What am I doing wrong?

Thanks for any help.