Under the Where you're logged in section, tap on ⋮ button next to the entry showing the "Messenger" option.
Scroll down and choose Security and login.
In the social media app, follow these steps to log out of the Messenger application: If you've got Messenger on your smartphone, chances are pretty high that you also have the Facebook app installed as well. Log out of the Messenger session from the Facebook app (Android and iOS users) Let us now take a look at each of these steps. The first procedure works on both iPhone and Android phones, while the second is only available for the Android Messenger app. There are two methods of logging out of the Facebook Messenger app: Log out of the Messenger session from the Facebook app or clear Facebook Messenger app data from within your device Settings.
How to log out of Messenger on Android and iOS
How to log out of the Messenger app from the Facebook website.