Why doesn't the Huntress card not have her real identity before her character name? She WAS BATGIRL after all. I thought the whole point of character identities was to handle the problem of character's being more than one superhero and to deal with the fact that more than one person was the same character.

John Paul Valley was Batman 500, as well, but Azrael doesn't have an identity.
Some characters, I assume, just aren't going to be made, like Barbara Gordon <> Batgirl.

Of course they're going to make Baraba Gordon <> Batgirl.

I'm sure we'll see her in the Superman set. Just you wait.

So, just how DOES DC handle the uniqueness. Is it by either name, or both, or just the actual name, or what?

Uniqueness is checked by anything before the <> in the card name.

Dick Grayson<>Nightwing
Dick Grayson<>Robin

Both cards would be considered Dick Grayson. Even though one says Robin and one says Nightwing only one of the two could be out at the same time. Either one could be used to power up though.

Huntress was Batgirl?

All I know of Huntress i courteousy of the WB and Birds of Prey...

Yeah, Huntress was the "new" Batgirl (the one with the closed-face cowl) before Cassandra was, for a couple of issues during No Man's Land.