You may be able to get what you are wanting from the KML reader if you expose more of the kml format attributes (there are a lot). For folder names and identifying what folders they are in I believe the ones that you want are kml_parent, kml_id, and kml_name.
When you read in your KMZ you can choose to between individual feature types or merged. Either will work but if you choose individual then you will get a Reader FeatureType called Folders separate from your data. Once you read in the data, the kml_name attribute is the display name of the folder and the kml_id is the id of that feature/folder (could be the same as the name but may not be). The kml_parent is the kml_id of the folder that each folder or feature is in.
This guide is for creating kml folder structure in fme, but it may help explain how the attributes are working: https://support.safe.com/hc/en-us/articles/25407754952717-Creating-KML-Folders-Through-KML-IDs