I started to give a flippant response but then remembered all the discussions on here about how some other words are defined by some people. A lot of people don’t use the dictionary definition of a word which can lead to a lot of confusion.
There are different “game” breeds. These breeds are recognized and are certainly not crosses.
“Game” can also refer to a type of chicken, usually a fairly small and wiry bird. When people ask what breed their chicken is you’ll sometimes see people say “It looks like he has some game in him”. That’s more of the type than a specific game breed.
It’s possible that person simply means they are not purebred chickens when he calls them games, but if you cross an Australorp with a Rock, or a Wyandotte with an Orpington, you do not have a purebred game breed and you do not have a chicken with “game” characteristics.
Edited to add:
That kind of reminds me of when I was working with the Boy Scouts. After you try to teach a skill and the nomenclature a boy would occasionally say "Well, I can call it that, can't I" My response was "Yes, you can, but you would be wrong".