**Ditch the `Enum`.**

[![enter image description here][1]][1]


AFAIC Using an enum property to list the materials has been superseded by using a pointer property to the ID object type. In this case a material.

https://blender.stackexchange.com/questions/199233/custom-search-data-to-uilayout-prop-search

https://blender.stackexchange.com/questions/74904/search-custom-properties-in-objects-on-the-scene-python/101303#101303

Have used this option here.  Given the property group a `material` property, which is a pointer to blend material, or `None` if not set.

An update method on this property sets the objects active material to the sauna specific one chosen.

Added a boolean property to material type to flag (poll)) it as a "Sauna Cloud" material.  The only materials that poll  will have this set as true.

A select menu is drawn by the UI using a `UILayout.prop(...)` (commented out) or with the ID template.  (_Note_ a custom operator to  add material can be set here, I took out the generic add material_) 

**Other alterations.**

Drew the context objects material slots in the  panel and an option to add a new one. Further to this would be to define your own `UIList` to add the "Is sauna material" checkbox in the list, add a filter etc.

Displayed the is sauna material checkbox, in panel and prepended it to Material Preview panel, by way of example.

Set the context members `material` (_`context.material`  only exists in the PROPERTIES area, otherwise needs to be set for material operators_) which can used in your addons custom operators, with buttons in this layout,  as

    mat = context.material
    sc_mat = context.sauna_cloud_material

or in hindsight  `sc = context.sauna_cloud`. 

Changed the namespace from `SC` to `SAUNA` and would go as far as to say even `SAUNA_CLOUD`  The naming convention,  `_PT_` for panel type.  is all about limiting name clashes. 

    import bpy
    from bpy.props import PointerProperty, BoolProperty
    from bpy.types import PropertyGroup

    class SaunaCloudPanel:
        bl_space_type = 'VIEW_3D'
        bl_region_type = 'UI'
        bl_category = "Sauna"
        @classmethod
        def poll(cls, context):
            ao = context.active_object
            return ao and ao.type == 'MESH'

    class SAUNA_PT_Wall(SaunaCloudPanel, bpy.types.Panel):
        bl_idname = "SAUNA_PT_Wall"
        bl_label = "Wall"

        def draw(self, context):
        
            layout = self.layout
            sc = context.scene.saunacloud
            ob = context.active_object
            mat = ob.active_material
            layout.context_pointer_set("material", mat)
            layout.context_pointer_set("sauna_cloud_material", sc.material)
            box = layout.box()
            box.prop(ob, "name")
            col = box.column()
            col.template_list("MATERIAL_UL_matslots", "", ob, "material_slots", ob, "active_material_index", rows=5)
            col.operator("object.material_slot_add")
            if mat:
                col.prop(mat, "is_sauna_cloud")
            
            
            #col.prop(sc,  'material', text=getattr(sc.material, "name", "None"))
            col.template_ID(sc, "material")
            


    class SAUNA_Global_Props(PropertyGroup):
        def poll_material(self, material):
            return material.is_sauna_cloud


        def update_material(self, context):
            scene = self.id_data
            sc = self
            ob = context.object
            mat = self.material
            # debug print
            print(f"Update object: {ob.name} with material: {mat.name}")
            ob.active_material = mat
            


        material : PointerProperty(
            type=bpy.types.Material,
            poll=poll_material,
            update=update_material,
            )          

    classes = (
                SAUNA_Global_Props,
                SAUNA_PT_Wall,
            )

    def draw_sc_material_checkbox(self, context):
        mat = getattr(context, "material", None)
        self.layout.label(text=mat.name if mat else "None")
        if mat:
            self.layout.prop(mat, "is_sauna_cloud")

    def register():
        for cls in classes:
            bpy.utils.register_class(cls)
        bpy.types.Scene.saunacloud = bpy.props.PointerProperty(type=SAUNA_Global_Props)
        bpy.types.Material.is_sauna_cloud = BoolProperty()
        bpy.types.MATERIAL_PT_preview.prepend(draw_sc_material_checkbox)

    def unregister():
        for cls in classes:
            bpy.utils.unregister_class(cls)
        bpy.types.MATERIAL_PT_preview.remove(draw_sc_material_checkbox)

    if __name__ == "__main__":
        register()


**To answer the question,** in an update or items method `self` is the propertygroup and `self.id_data` the ID object down the chain it belongs to. Scene in this case.

To get the enum value in either

    self.matlist


Consider defining an enum and using same on material and object to poll instead of relying on naming convention. 

    enum = EnumProperty(... our types ...)

    bpy.types.Material.sauna_type = enum
    bpy.types.Object.sauna_type = enum

You can't set properties of ID objects in a draw method.  

In the update method, to do same as above

    ob.active_material = bpy.data.materials.get(sc.matname)

however IMO far better to poll a pointer. Import `from bpy import context` into the poll method to get context object. eg using matching enums, 

    def material_poll(self, material):
        from bpy import context
        ob = context.object
        return material.sauna_type == getattr(ob, "sauna_type", None) 

or via name slicing method used in question

  [1]: https://i.sstatic.net/nRlCW.gif