I have also noticed I don't receive the notification on some text messages. When the notification doesn't come through, my apple watch does not get the text message either. Very annoying when my phone is in the other room and I'm trying to log into a website that has two factor authentication. I haven't found a permanent fix but hoping the info below helps apple tech troubleshoot. I'm in the tech field so I tried way too long and documented what I tried (NERD ALERT)! NOTE: I ALWAYS have my phone on silent mode, but when text notifications work, I receive a short vibration (and my apple watch gets the message too. When I don't get the text vibration notification, I don't get it on my watch either). I just purchased my iPhone 12 mini and new watch series 5 last week and I'm running software version 14.3. Below is everything I tried.
Today, I was consistently not receiving my 2 factor messages from a specific short code number (happened to be USAA). I was trying the same things all the other "fixes" said to try and nothing worked. I finally did the following and it worked temporarily (at least 2 times consistently, then stopped again). Unfortunately, this "fix" would have to be done for every contact you want to ensure you get the notification from.
After trial and error, I had already created a contact called "USAA 2 factor", so I clicked on info-->edit-->text tone-->vibration and changed it to "quick" (before it was set to "Staccato (Default)". I tried it again and BAM, received the notification and it went to my watch)!
It worked three times without a problem, then quit working again. Then I set up the contact for work short code two factor, but I can't remember if the adjustment worked on tat one (I don't think it did). So i reset my network settings (settings-->general-->reset-->reset network settings). Once complete, the texts worked three times in a row again. Then stopped again. I also set up a contact for my work two factor account and tested it. The first text notification came through. Subsequent ones didn't. (so the first 3 USAA notifications came through, two more USAA notifications didn't come through. Next I tried work, first one came through, two more did not. So I reset my network settings again. This time I was able to get two work notifications, nothing on the third one, nothing on the fourth one. Went back to try USAA, nothing on the first one. I finally give up.
Also, I was thinking this was only related to "short code" text messages but my father texted while I was waiting for the notification of one of my two factor attempts....NO Notification!!!
Seems to have something to do with the network settings...i tried over wifi and over cellular and got the same results each time. Notifications definitely worked for me for a short period of time afterward. Not sure if it's a count of messages, count of messages from the same person, length of time after network reset or intermittent network something.
So frustrating!!!! Fix it!