I think the thing that everyone has to realize is that the AI in your game is NOT sapient, self-aware, or anything really. It's a programmed set of instructions. A Strong AI is an AI that can actually do something that the original programming planned for, it can learn. The AI in your game can learn, sure, but it's only in that specific area. When you shut it down, those are erased. It can't even properly be called an AI, it has no intelligence, it has a preprogrammed set of actions and reactions. The AI that we should be talking about is a Strong AI, which is also known as "artificial general intelligence". The key word here is "general". An AI in a game is specifically meant for that game, and doesn't even come close to a Strong AI, no matter how good it is.
There isn't even much that a Strong AI can do, that a regular old AI can do. Almost any purpose you can think of, you can program an AI for, even a very vague area. A Strong AI is actually virtually useless in the category of doing any tasks. The only reason I can think of to create one is for philosophy and things related to it. Otherwise, we have zero reason to even build one in the first place. Unless we wanted to "upload" your brain or something of course, but that's not definitely possible. Even then, it's just the empty shell of an AI, built as a replacement for your brain.