You can completely disable desktop notifications or you can set option per web site. Go to settings and search for notifications (or whatever is in your language) and then manage exceptions.

I think you didn't get it. Notification windows's space is not enough sometimes for whole message, but now there's additional space occupied by the website address. well, now I don't have any other sites which provide desktop notifications, so I can't check whether it's's feature or Opera's one, that's why I started this topic. I can only say that two weeks earlier there was no address in notification