The Application Data folder should be called manager.io such as /Users/caprat/Documents/Manager.io You can check on the left-bottom of your screen if that is indeed the folder your Manager app is using.
Copy from where? How do you make these backups and where do you save the initial ones?
Please confirm hat this is the Desktop and not the Server edition.
What operating system (such as Sonoma 14.5) are you using on your Imac ?
What processor Intel or Apple silicon (M1, M2 or M2 also referred to as ARM) and is it 64 bit? The download page on manager.io only gives the dmg for the Intel version while ARM-64 for Mac can be downloaded from https://github.com/Manager-io/Manager
This will NOT resolve any of the reported data issue but ensures that you use the right Manager app installation.