I recently purchased a photography plan (LR and PS).
and Im using a MAC El Capitan.
When I open the app, still on trial and adobe cannot detect I already purchased.
Have you download and installed the Creative Cloud Desktop application? If not, you need to do that as it's this app that controls the licensing activation of the photography plan apps. Once installed, log in to your Adobe account, and that should sort out the Lightroom problem. If it does not, you might have to uninstall Lightroom, then use the CC desktop app to install LR and PS.
Try uninstalling Lightroom, then in the CC desktop app try signing out and back in again (in Preferences, click on the gear icon). Hopefully that'll shift Lightroom into the "Install" state, as PS already is, then click on the Install button and the app will take care of business.