No. The Dark World originated in the backstory to A Link to the Past, as Ganondorf touched the Triforce and wished for the land, changing the Golden Land (Sacred Realm) into a mirrored world of the Light World. It still reflected the heart, though, as it's the Sacred Realm, so all those who enter turn into what they are inside their hearts.
The Twilight Realm is something different entirely. In fact, it's actually much different than the Twilight that covered Hyrule in the first half of the game. That Twilight was unnatural to the Light World, so it somehow turned people into spirit forms. Link, due to the Triforce of Courage, didn't turn into a spirit form, but rather what his spirit was
like. A feral beast. A wolf. Now, the Twilight Realm is just a realm with the absence of light entirely, and those who enter it change into shadows of the light, which are known as the Twili. Link didn't change because of the ToC.
Now, the Siren World we really have no idea on. All we know is that it (appears to be) a mirrored world of Hyrule, and that we will be collecting things similar to the Tears of Light in Twilight Princess while avoiding the guardians of the Siren World. We won't have a sword (or shield, I think), so we will have to use stealth, and if Link is touched by one of the guardians, he's sent out of the Siren World.
(Sequence photo from Djinn's article about the Siren World)
