I was having this same exact problem on my 6S Plus running the latest iOS and Facebook and Messenger apps, it started some weeks ago. I was going crazy trying to fix it and the way I got it working sounds crazy.
Backstory and instructions: I got tired of asking friends and family to message me to see if it was working after changing every setting so I created a fake Facebook account and I just added myself as a friend to test it myself. Before I sent a test message to myself I enabled all notifications for the Facebook app, I had them disabled in Settings and I just allowed Messenger notifications. I sent my self a message on the fake account I just made and all of a sudden everything was working again, sounds, badges, banners. It seems crazy to think that adding a new friend can fix it but for me it did, and I tried everything prior, uninstalling Facebook and Messenger, nothing worked. I than did one more test and closed all apps and restarted the iPhone by holding down the power button and home button to see if it would still work after a reset and it did, I than had my friends message me and it worked, when it wasn't working just 30 minutes before I added my test account as a friend. I than disabled Facebook notifications again in settings and tested one more time and it still worked.
Site note: My fake Facebook friend was from Reykjavik, Iceland and is attending Háskóli Íslands college beginning in 2012 with the birth date of September, 23, 1992. I doubt that has anything to do with it but the fix was crazy and maybe Iceland is good luck lol.
