I think I know when it appeared to be happening. I had recieved some SMS text message from my bank regarding transaction item. And there was some special weird symbol, which looks like vise versa question mark.
I had something with macOS Notes, when I opened it and it just crashed. So I just cleaned all content in Notes in iCloud web. And was able to use macOS Notes again.
Is there any way I can clean all text messages in Messages app on iPhone?
But it keeps closing in right after I open it to write someone a reply. So basically I just can view notification previews and can't send a reply to anyone.
I have in mind some ways of fixing it. Like restore/refresh or wait for next iOS version to fix it or refresh it somehow.
Though, I don't want to restore the phone, settings or whatever. Might just probably wait when it cleans messages older than 30days of new iOS version will come up.