I partially agree with EdwinAshworth.
We routinely say " is today". So the normal phrasing would be, "The game between A and B is today" or "My birthday is today".
It can be said the other way around, especially as an exclamation. "Today is my birthday." "Today is the big game!"
But putting "today" first is generally only used when you want to emphasize that it is today, as opposed to some other day. The "normal" word order is to say " is ". In English, changing word order is often used for empahsis. In this case, it emphasizes the day. So if a person says something like "Today is the big game!" he probably means that he is excited because the day has finally arrived after a long wait.
Note all of this applies to dates other than today. "My birthday is next Friday." "Next Friday is my birthday!" etc.
The same applies if you turn it into a question. The normal phrasing would be, "Is the game today?" But if you want to emphasize the date, you could ask, "Is today the game?"
But when we are asking about a category of event rather than a specific event, we add the word "there" to a question. That is, we might ask, "Is the game between A and B today?" But we DON'T ask, "Is any game today?" but instead "Is there any game today?" Not "Are company holidays in November?" but "Are there company holidays in November?" (Perhaps someone else can give a good reason for this. I don't know why: we just do it.)
RE "Is it your birthday?" The day is implied. If no date has previously been mentioned, the hearer would normally assume you mean "today". If you were just talking about some other date, they would assume you meant that date. Like, Al: "I won't be here on Thursday." Bob: "Is it your birthday?" Al would assume that Bob meant "Is your birthday on Thursday?", i.e. is that the reason why you will not be here.
Yogi Berra, a famous baseball manager, was once asked, "What time is it?" and he replied, "Do you mean right now?"