No, I don't think that should still be showing up there. That seems to be in the same place as where it tells you when mods on mode is active - might be interesting to try changing your settings to mods on and starting the game, and see if it has any effect (don't know why it would, but then again don't know why it's still saying demo mode either).
Have you bought any DLC (planes, maps etc.) yet? If not, maybe it would be worth creating a new account with a different e-mail address (you can create and use a free g-mail acct if you don't have another one to use), and seeing if that demo mode thing is still there when you start the game with your new account.