However, when I drag and drop it into Unity, hoping everything would just work, my asset is just white/gray. I also noticed that the mesh looks strange in Unity, see photo.
Would you have a screenshot of the Material panel from the inspector when you select that model?
Some clarifications that may help:
If your colors are set using textures, looking at your blender settings, they should be embedded in your FBX file, which means they have to be extracted from the FBX once in Unity (because the FBX importer doesn’t handle the textures, they have to be imported on the side).
In order to extract them, you should be able to click on the “Extract Textures” button from the Material panel of the inspector. It should create the textures in your project and automatically re-import the fbx file using the textures.