Your account won't go anywhere, even if you delete the game from your computer. I'm not quite sure what you want to back up. All of your beatmaps, skins, etc, are all stored locally.
Right click on osu! and hit "Show Package Contents" then go to "Contents > Resources > drive_c > osu!" in this folder you will find your Skins folder, Songs folder, Data folder and a bunch of different db files and cfg files. Depending on what exactly you want to save, the files and folders you need to backup will change.
Also in your osu! folder is a file named "osu!.pcname.cfg" where pcname is the name of the user account on your computer. Edit this file in any text editor (TextEdit is installed by default on macOS) and change the line "CompatibilityContext" to 0. Save the file and compatibility mode should be disabled.