After "upgrading" to the latest Remote Desktop app on macOS -- now called "Windows App", you will not be able to save passwords as the app doesn't have access to your keychain for previously-stored passwords. To remedy this:
- Open Keychain Access
- Open the default keychain where your passwords are stored (usually called "login").
- Search in the upper right for "com.microsoft.rdc.macos" -- this will show the stored password items.
- For each item:
- Right click and select Get Info
- Click Access Control tab
- Click +
- Navigate to Applications and select Windows App
- Click Save Changes - this will require authentication.