Well, your analogy in the second paragraph implies theism or deism. If mathematics is game-like because it's a description of an actual game that's governed by rules or laws, then the external game must have a lawgiver or an inventor. That's the religious answer to the question of why there are laws of nature at all.

I think mathematics is game-like just because that's our epistemic strategy in trying to understand something that isn't really a game. If we assume atheism, there are no natural laws or rules in any literal sense. There's a natural order that unfolds propabilistically, as modelled by scientists and so on, but those regularities aren't artificial, intelligently designed, or ruled by any imperative. That makes nature monstrous (zombie-like).

In short, I think mathematicians play games partly to distract from the horrific existential upshot of scientific naturalism.