I just had a thought - for those getting the ad's, is your Origin in-game overlay enabled or disabled?
I've had the Origin in-game overlay disabled since I got the game (disabling it in DAI gained me 5-10 fps in the benchmark). I'm wondering if that is what is causing you to get the ads while I'm not? Since obviously it has nothing to do with actually doing any multiplayer.
To test, if your Origin in-game overlay is on... right-click DAI in Origin, go to "Game Settings" and tick the box "Disable Origin overlay" - once disabled, start the game and see if your still getting the advertisement or not.
Origin in-game overlay might also explain why the advertisements look like something stuck on the screen - because that's basically what it is (if it's the overlay causing it).