Is this another way for saying "good morning"? What countries use this greeting regularly?
Yes, it could be another way of saying "good morning", but generally speaking it means more of a "how's it going?" or "how are you doing?" when you see someone in the morning.
It's a fairly casual way of greeting friends. A variation is "¿Qué tal amaneciste?"
It also means "How did you sleep?" or "How did you feel waking up?" I would say, it is a question that implies more than a greeting. It's like the person asking wants to know if you have rested well. I'm not sure which countries uses this phrase more regularly, but I've heard it in Mexico for sure.
Here in beautiful Colombia, it means "good morning, how are you?" We also say "como amaneció," which is to see the sky break. One friend amended this to "como amanicielo," which I believe is, "good morning how are you my sky?" a bit more affectionate, sky being a word of endearment. ¡Love Spanish!
Another way this phrase is used is, ¿Cómo te amaneció? (How did your day begin?)