Leaving aside whether Facebook's app genuinely does this (which is a separate question), is this possible? I had always naively assumed that an app couldn't access what was on my clipboard unless I explicitly chose to "Paste" into a native text view. Is that assumption wrong? What is the security model for the content of the clipboard on the two major phone OSes? (Or what are the security models, if it's handled differently between iOS and Android?)
Prior to Android 10 any app could freely register listeners to receive the clipboard contents whenever they changed.
As of Android 10, only the current app with focus and any app set as the input method editor (i.e. the keyboard app) can read the clipboard. Previously common methods of creating background services that would listen to changes to the clipboard are no longer possible and background applications that used to rely on clipboard data must now implement workarounds where they receive the focus at least momentarily before they are able to read the clipboard.
Additionally in Android 12 or later it is possible to enable a feature which alerts the user (in the form of a toast) whenever clipboard content is read by an app.
As confirmed in the comments by user 11684, even Apple allows apps to read your clipboard (though only while they are in the foreground). Here is the link to the documentation that returns the data of the clipboard.
Update in iOS 14: Beta version of iOS 14, reportedly, has a feature which alerts the users whenever any app reads the data from the clipboard.
It is definitely possible for an app to check the clipboard content anytime.
Example: copy a link, then open Chrome. It will offer the link in the search bar (‘goto link you copied’). If you copy something that is not a link, it will not make that offer. This proves that it checks the clipboard without explicit user action.
In iOS 14, iOS now displays a system notification alert whenever an app reads from the clipboard.
App1 pasted from App2
This caused many privacy-focused stories to be written in the summer of 2020 as various apps were "caught" reading clipboard data during the iOS 14 beta: