Facebook’s Messenger and Facebook applications can experience sign-in issues if you have the Firewall ”Lockdown” installed.
If you’re familiar with this application or need a reminder: https://apps.apple.com/us/app/lockdown-privacy/id1469783711
Is a link to the application in question.
This application is a safe one with multiple testings as shown here: https://openaudit.com/lockdownprivacy
However, this application does have issues with the trackers it blocks also preventing you from signing into Facebook on your mobile device.
So when opening the application there will be a ”power” icon displayed with a circle and a line intersecting it. Just pressing this button will turn it off. Return to your Facebook application and sign in. Know that after you sign in you can reenable the security of Lockdown and it will not sign you out.