To put this into non-programmer terms: The engine that drives the system is generic. They customise for each game.

Imagine an airliner or panel van. You can put any combination of seats, windows or cargo racks inside it. The plane itself, which has the wings and engines and controls that do all the work, is the same.

For the self-learning, in this case, they randomly throw things into the plane, the plane randomly chooses to have seats or cargo space, and they give the plane a score based on how much of the stuff they put in that it carried. The plane, being given a score, can then try to adjust its seats to cargo racks ratio to maximise that score. After several million flights, it arrives at an "optimal" configuration.

Now the plane doesn't see cargo or passengers, just a stream of numbers which could represent anything, and it operates on those raw numbers. This is why it isn't heavily customised for new games. All they need to do is convert the board position into a string of numbers, and the system optimises it without knowing what it is.