Ok, finally got it to work. The documentation for Use Messages in iCloud mentions below (emphasis mine).
To turn on Messages in iCloud, make sure that you've set up iCloud and
are signed in with the same Apple ID on all your devices. You also
need to use two-factor authentication with your Apple ID, and turn on
I already had two-factor ...
I ran into the same issue with macOS 11.0 and I am not sure if it happened because of Big Sur using the new Messages app. What resolved this was to go to my Preferences > iMessage tab and select the Enable Messages in iCloud option by checking the checkbox. Then, click the button that says sync now and they will sync.
Hope this helps!
By experience, latest macOS (11.1) and iPadOS (14.4) versions let you get a new line by pressing the Shift ⇧ + Enter ⏎
In older versions of iPadOS it would be pressing Option ⌥ + Enter ⏎ (same as for macOS)
Here is the method I have used to extract my iMessage History (using a MacBook).
Find your chat history database chat.db. It is located at ~Library/Messages which is accessible from Finder.
Use a Database Management System to extract the info you need. If you use MacOS, sqlite3 is preinstalled in your system.
Open the Database using sqlite3 ~/Library/...