this panel should be located in the Scene Properties tab

Here's the updated code to make the panel located in the Scene Properties tab:

import bpy

class AssetCollectorPanel(bpy.types.Panel):
    bl_label = "Asset Collector"
    bl_idname = "OBJECT_PT_asset_collector"
    bl_space_type = 'PROPERTIES'
    bl_region_type = 'WINDOW'
    bl_context = "scene"
    bl_category = "Asset Collector"

    def draw(self, context):
        layout = self.layout
        row = layout.row()
        row.label(text="Assets folder name")
        row.prop(context.scene, "asset_folder_name", text="")
        row = layout.row()
        row.operator("asset.collect", text="Collect")

class AssetCollectOperator(bpy.types.Operator):
    bl_idname = "asset.collect"
    bl_label = "Collect Assets"

    def execute(self, context):
        print("Collected!")
        return {'FINISHED'}

classes = [    AssetCollectorPanel,    AssetCollectOperator]

def register():
    bpy.types.Scene.asset_folder_name = bpy.props.StringProperty(
        name="Assets folder name",
        default="assets"
    )
    for cls in classes:
        bpy.utils.register_class(cls)

def unregister():
    for cls in classes:
        bpy.utils.unregister_class(cls)
    del bpy.types.Scene.asset_folder_name

if __name__ == "__main__":
    register()