Actually, contrary to what a lot of people seem to assume, I think the Kingdom Hearts in the first game was no the door we saw. If you look closely, there's a giant heart-shape in the darkness behind the door, meaning that the door we saw was actually what it said to lead TO Kingdom Hearts, the Door to Darkness, which is why, when the door was opened, we saw INTO Kingdom Hearts, not that the door WAS Kingdom Hearts.
Anywho, as far as the "different forms" thing goes, I say yes. I don't there there has to be just ONE Kingdom Hearts, it's just that one is more powerful than the others.
The definition of a Kingdom Hearts is just a massive collection of a certain type of heart. The Kingdom Hearts that Xehanort's Heartless was after is the most powerful form, the Kingdom Hearts of Worlds.
The Kingdom Hearts made by the Organization was a Kingdom Hearts made of Hearts. Less powerful, but it wold give the Nobodies what they wanted: hearts. Hearts to become whole.
As far as we know, there are no other type of hearts, but it's possible there's some other version of a Kingdom Hearts. Unlikely, though.
It's also possible there are more copies of a Kingdom Hearts of Hearts. It's not like the one made by the Organization was made of every heart ever lost to darkness. Meaning they had to go SOMEWHERE, right? This might also mean there is more than one Door to Darkness, because there also should be more than one End of the World somewhere, and if not, then another Door to Darkness coul have made itself, or have been made, somewhere, most likely at the End of the World.