Hi--I am not an Apple expert. Try opening each app on your new phone and logging in. Then reset Bluetooth connection a couple of times. Then restart your Iphone once or twice.
Here's why that fixes some notification problems: your new Iphone is NOT automatically connecting to the apps with your log-in info (some of them accept face recognition so you have to go thru the steps on each app). You can see the apps and see the settings, but you can't tell they are NOT logged in on your new phone. Turning Bluetooth on & off and restart just clears out old settings.
It worked for me--I had same thing as you a few days ago--now it all works fine. Maybe Apple hasn't done a very good job telling new owners what steps they have to take with a new phone. Like many, I upgraded from Iphone 6--an ancient device that I knew very well. I had to figure the rest out myself. Hope this helps you.