EA accounts and Origin accounts are separate accounts, at least for any the older EA games that are installed from DVD-ROM. For example, I have Undercover, Shift 1, and Hot Pursuit (2010), all from EA DVD-ROMs, on my EA account, but none of those games show up in my Origin account. One of my friends bought the DVD-ROM version of Shift 2 Unleashed Limited Edition, and that does not show up on his Origin account. I bought Shift 2 Unleashed Limited Edition download version from the EA Store, and it now shows up in my Origin game library, but the Origin version is different, than the EA Store or DVD-ROM versions. The EA Store download and DVD-ROM versions appear to be the same version, only the Origin version is different. The Origin versions of these games will launch the Origin app, and can only be run from the Origin app, while the EA Store or DVD-ROM versions don't need the Origin app (Origin didn't exist at the time these games were released).
You can install, and activate (activation is handled by solidsheild.com) Hot Pursuit (2010), without ever logging in and the game will not be associated with any EA account. If you do log in, you log into an EA account, and as mentioned above, the game will not appear on your Origin account. It's possible that the activation key can be used to add the game to a "different" Origin account, since there seems to be disconnect between DVD-ROM versions and Origin versions of the same game, along with issues of players creating Origin accounts with different emails during the transition from EA Store to Origin.
You can contact EA support to see if they can fix this.
Side note - I have read about issues with Origin download versions of games. The main one I recall is with Undercover. For the DVD-ROM version of Undercover, there's a patch 1.0.1.18 that adds the Challenge Series, where a player can win 3 special tier 1 cars. Players have posted that for the Origin version of Undercover, the Challenge Series is either completely missing, or is not working (the screen shows up , but you can't start the Challenge Series). Some Origin games used securom server for online authentication, and that server has been shutdown, rendering the affected games unplayable, except for a few where it was possible to disable the authorization.
