If you are editing the Mesh
resource, you should see in the Inspector Panel a series of properties named "Surface 1", "Surface 2" and so on. Each one has a material assigned. There you can replace the material with your new ShaderMaterial
(alternatively you could edit the materials that were imported along with the model).
Similarly, if you are editing a MeshIntance
, you should se a series of material properties which are overrides for the materials of the Mesh
resource, and you can also se the ShaderMaterial
.
If you don't see multiple surfaces/materials, then your model got a single surface. And no, Godot does not have the means to divide surfaces.
Addendum post edit: Yep, that is a signle material in your MeshInstance
. Multiple look like this:

And in the Mesh
they look like this:

You can open the Mesh
resource from the MeshInstance
by clicking on its Mesh property, it should be an ArrayMesh
.
As you have certainly found out, you are not going going to get complex Blender materials on Godot. Both parts have a bit of blame here, because Blender materials are not standard, and only the simplest can be encoded in standard formats. If you are using a Godot exporter addon in Blender, it might try its best to translate the materials, but some just cannot be recreated, and that part has to do with Godot limitations.
Thus, my advice is to set simple materials for the different surfaces you need. And then if you need more complex materials, override them in Godot with materials that work on engine.