How common is it for people to not have the concept of certain ‘breakfast food’? I recently visited Ukraine with a friend. One night we stayed at a hotel that served meals. In the morning, we came down to breakfast and were surprised to be served a carbonara-type pasta dish–not what we would normally have considered breakfast food. We asked our guide, and he said that Ukrainians don't separate foods into different meals and just eat whatever, whenever.

How common is it for people to not have the concept of certain ‘breakfast food’?

This article goes through common breakfasts from different countries, but most seem to be uniquely breakfast-y (except for Korea).

This one claims that the US is in the minority in having unique breakfast foods and suggests that ‘poorer people everywhere, especially in places like India and China, eat the same kind of food for meal after meal’.

Ukraine doesn't get a mention on the Wikipedia article on ‘Breakfast’, so I can't even verify what the guide said about it–maybe it was just the hotel we were in's approach–to be fair, the hotel was in the Chernobyl Exclusion Zone so I can imagine food options might be limited.
In Japan the word for rice is gohan, breakfast is asa gohan, meaning "morning rice". Dinner is ban gohan. You get the idea - rice is the traditional part of every meal. 

Traditional Japanese breakfast food would be rice, fish, miso soup, vegetables, pickles. Which is a lot like a traditional lunch. More recently western breakfasts foods have been taken up some, so the concept of separate breakfast food is more of a thing.