I too wasn't getting banner/popup notifications for all of the things I used to before switching to the new Teams. I was still getting some and couldn't figure out why. I even checked my settings several times. But just now, I finally realized that my there was a change in one of the notifications settings from the old to new Teams.
On the old Teams, it was Settings > Notifications. Under "Teams and channels", you'd select the option you wanted.
In the new Teams, it's Settings > Notifications and activity. Under "Posts in shown and pinned channels", it was marked off as "Off". After switching it to "Show in Activity and banner", I'm now getting all of my banner/popup notifications again.
Note: I declined to get notified about responses to my post.