Which recording app are you using? In most cases, the Overwolf recording apps such as Outplayed have several settings options:

1. A setting to automatically record the game. These types of settings rely on the recording app recognizing the game process, if more then one game process is running at the same time this can cause issues.

2. A setting to manually record the full screen. As these types of settings record the full screen and not a specific game process you can use this setting if you would like to record several games at the same time.

Hope this clears things up!