I have not finished playing through either, but I find myself wondering if the chosen undead is supposed to make it through all of Dark Souls, and be the main character in Dark Souls 2? And is the chosen undead in Demon Souls as well?
To begin with, as others have mentioned, Demon's Souls is a whole other universe, completely separate from the Dark Souls universe.
Since no one has come up with any hard evidence for either case, this is all speculation, so I'll give evidence for both cases:
- The Chosen Undead is the same being
"Time is convoluted..." This phrase is used constantly throughout the Dark Souls Lore. This explains that Heroes existing in multiple universes (multiplayer, as well as other characters you come across, such as Iron Tarkus and Solaire) exist in their own time lines, and can cross over into others. There is nothing stopping us from saying that even though the time frame between Dark Souls 1 and Dark Souls 2 is immense (an untold period of time has passed between the events of Dark Souls 1 and the Beginning of Dark Souls 2), that the hero has not survived by slipping in and out of the timeline. Even at the beginning of Dark Souls 2, the hero enters a portal, transporting him through time.
- The Chosen Undead is a different being
This is the more likely case, as even though the Chosen Undead was an incredibly powerful being - having absorbed the Souls of the Gods, at some stage in the past, the Flame was re-kindled. This can be proven by the fact that to re-kindle the flame in Dark Souls 1, the Chosen Undead sacrifices himself, and the Souls he has collected. This can be proven again by the fact that you can once again collect these Souls. Eg; the four "Old Souls" (Old King Soul, Old With Soul, Old Dead One's Soul, Old Paledrake Soul) description reads:
- "This once-magnificent soul continues to exert influence over the land, even after the eons have reduced it to these remnants."
This is proof that these Souls have been released from the original vessel - the Chosen Undead. But whether he chose to sacrifice himself, or he was killed by another, and they used these souls to re-kindle the flame, we don't know.
Edit: With the release of Dark Souls 3, the storyline, lore, and the environment show that the First Flame was linked at some stage in the past.
Some people refer to DkS3 as "DkS1.5"
Which again adds extra grounds to the argument that the Chosen Undead is again a different being in each installment of the game.