But this is the specific game we have today at some specific place at some specific time. And there is only one game that I am talking about. So why is it not 'we have the game today?'
It is a common misconception that the indefinite article cannot or does not refer to a specific referent. In fact, it can refer to either a specific referent or a non-specific referent. What the definite article does that the indefinite article does not is refer to a definite referent. We are, after all, talking about the indefinite and definite articles here, not the "inspecific" and "specific articles."
I want to marry an English woman.
An here refers to a non-specific English woman. It's kind of like saying any English woman, and I don't care which one, just as long as she is English. Now that is not specific, is it?
But if you say
I married an English woman, and her name is Jane.
Here an refers to a specific English woman. But since you have used the indefinite article, you are not marking the noun phrase English woman for definiteness.
If you say
I married the English woman
you are marking the noun phrase English woman as definite. What this very often means is that you expect that your addressee (the person you address the sentence to) can identify which English woman you are talking about.
We have a game every week
you are not talking about a specific game. But
We have a game today
is talking a specific game. What it is not doing is marking this specific game for definiteness.
If you say
We have the game today
then you are signalling to your addressee that he should be able to make some connection to 'game', either that he can identify which game you are talking about, or that he should consider 'game' as the topic about which you are going to start talking about.
The same can be said for the noun phrase specific game we have today. You can use a because the indefinite article can talk about specific referents. You can also use the if you want to mark the noun phrase as not only definite but also specific.