The waiting for "iCloud Logbook" seems to come to and end
Apple released with iOS 13 some new frameworks which make it now possible to sync your database across multiple devices by storing the database in iCloud. My tests so far were quite positive. Also several volunteer beta testers confirmed the functionallity works as expected. However there are some things to keep in mind.
What to consider
The technology behind the scene is the same used by Apple for synchronizing Notes, Photos, Contacts, Calendar, etc. via iCloud. So if this works for, most likely the logbook app synchronization will also make you happy. If those iOS feature already cause you trouble, you will obviously also not enjoy the logbook sync possibility.
You must be signed in to iCloud with the same Apple ID on both devices
Both devices must have internet access, and it can take up to 20 seconds until the data is synchronized to the other device
You can only have the entire database locally or synchronized in iCloud
To activate iCloud sync go to Settings > iCloud > and activate the toggle switch
After activating the iCloud database, the app is closed and when you open it again you will find an empty database, but this one is now synchronized into iCloud
If you deactivate iCloud in the settings again, you will have the local database from before
Note that the whole thing naturally requires more data volume
To import an existing (local) logbook into the iCloud database
Deactivate iCloud Sync (Settings> iCloud)
Export your database (Tools > Export > SQLite - You have to export the migrated database, the databases from the current App Store Release does not work)
Activate iCloud sync
Open the database you just exported with the logbook app and select a logbook to import
Give the device time and bandwidth to upload the logbook.
If you delete the app from your device, the data in iCloud remains. Once you reinstall the app, all data will be downloaded again after switching on iCloud support.
When activating or deactivating the iCloud function, the app has to be restarted, sometimes the change did not take effect and it is as if you had never pressed the switch under Settings > iCloud. Just try again