Did some more experimenting.
If you buy both a Steam licence or an EA app licence, it does not matter which order you purchase them, the EA app will priortize the EA licence first and will revert the games artwork back to the EA app version with no logo. This allows you to download the game exclusively through the EA app without Steam. However things change when you install the Steam version, if a Steam installation is present on your computer, the EA app will priortize the Steam version over the EA version. Once the EA app detects the Steam version, the games artwork changes to the artwork with the Steam logo; there is no way to revert to the EA version with the Steam version installed.
The EA app does not present you with a button to choose which version you wish to use to launch unfortunately. The only way to revert back to the EA version is to uninstall the Steam version and only have the EA app version installed. Also, you can not have two installations seperately, even if you have both the EA version or Steam version installed at the same time, no matter what .exe you start, be it in the steamapps folder or your EA folder directory, it will always run the Steam version as default.
So in conclusion, you can have multiple licenses, but Steam will be priortized over the EA app until you uninstall the Steam version in which it then reverts back to the EA version for you to freely use. You may also have to verify the download again, restart the app or clear the cache for it show properly in your installed category.
Hope this helps 👍