New iPad notifications still go to watch

I have bought new Air and I have found something that bothers me a great deal. When I was using FB on Mac and was chatting with someone, my phone/watch would not get new reply notifications from that person, which is logical and expected behavior.
Now when I am using Messenger on iPad, even in active conversation I get all received replies as new notifications on iPhone which then makes my watch vibrate like crazy. It makes using the iPad as Messenger chatting device a horrible experience.
Do I have something set up wrong, or is this normal for you guys, too? I have not yet been able to reproduce the same behavior for other apps but it might just be because I don’t get as many notifications as on Messenger.
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