It is normal to have the Files app not show anything. The purpose of the Files app is to store files that you want to use on another device. For example, since your wife has a Mac, anything you put into the Files app will be available on your wife's Mac on the iCloud Drive as long as the Apple ID is the same on both devices.
As for your iCloud already being full with the 5 GB data already, this would most likely be cause be Photos that were imported from her previous phone. There is a feature called iCloud Photos that will store the photos in iCloud and they will also be accessible on the Mac in the Photos app. Information about iCloud Photos is available here:
Set up and use iCloud Photos - Apple Support
If you do not use iCloud Photos, then your photos are backed up to iCloud in your Backup file. In either case, a large amount of photos will consume iCloud storage space. You do have the option to backup the iPhone to the Mac instead of using iCloud.
Back up iPhone - Apple Support
To see what data is being stored in iCloud, use this link to manage your data that is stored there.
Manage your iCloud storage - Apple Support
Sorry to hear about your struggles with transferring your data. There is no File Manager or File structure that you can access on the iPhone. The documents for the apps are accessible from the app itself. For example, to view your Photos, you will use the Photos app where you can edit or delete the photos. There is no File Structure to delete those photos from outside the app. This is different from the Files app, which as detailed before is used to sync the files to other devices and uses iCloud storage space to do that. In the Files app, you can use a file structure by adding Folders, but this is just like what you would find by using an External Drive.
Please ask if you have any questions of data that was transferred and you are unable to find on your iPhone. Of course, all of your Contacts should have transferred over and those will show in the Contacts app. Any apps that you had on the Android phone that is also available on the Apple App Store should have also downloaded automatically.
