Where are client settings and data saved?

If the client is installed, by default they are stored in the Application Data of the current user, standard behavior in each operating system.

Under Windows:

C:\Users\[yourname]\AppData\Local\Eddie

Under Linux:

~/.config/eddie

Under macOS:

~/.config/eddie

Portable mode, any platform

If the application is launched standalone (Portable mode), by default it saves data in the same directory of the executable file.
Only if the directory is not writable, it saves data in the user directory as explained above.
You can always control the path with the path options from a command line interface. The value can be a full path, or special values home (for the user directory) or program for the program directory.
Examples:
Eddie-UI.exe path=C:\mydata
Eddie-UI.exe path=home
Eddie-UI.exe path=program