I accidentally clicked on the "close" icon on the "open in app" banner on a Stack Exchange site, and now I can't get it back. As stated in this answer, it's permanently gone.
If you hit the × the banner goes away forever
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The sentence you quoted has a followup:
We store this in localStorage so if you clear your browser's memory we'll lose the setting, but for most people tapping it will mean never seeing the banner again.
So just clear the local storage of the browser app (your browser's memory) and you will see the banner again.