Wheee! Roll roll...
Hm, you turned into a rabbit, I see...
That's right, this world and the world where we live have very similar geography, but they're a weeee bit different.
On top of that, while they're lost here, people change into a form that reflects their heart.
I change my mind a lot, so I became like a ball...
But, if you have a sphere called the Moon Pearl, I hear you can return to your original form... roll roll roll
The Triforce appears to have ultimately been behind the changes.
I found the Moon Pearl!
It protects the Hero from the Golden Power that the evil one uses, and won't let his form change.
At first I thought that incidents like Link turing into a rabbit were just the luck of the draw. However, in TP, the form that seems to reflect Link's heart/nature is that of a wolf. Both ALttP and TP Link are caring, kind-hearted individuals who fight for the forces of good, and there is next to nothing that explains why one became a rabbit while another became a wolf. On closer observation, I found that other good people in the Dark World, such as the Flute Kid, mainly wound up with limiting forms as well. In contrast, wicked people apparently gained power in their altered states, the most notable example being Blind, who seemingly went from an ordinary thief to a powerful demon and dungeon boss in ALttP.
With this in mind, could the change produced by the Dark World be something intentionally put in place by Ganon to strengthen people who might potentially serve him while weakening those who might oppose him? Are the reflected forms seen in ALttP genuine, or is Ganon trying to ensure that, for example, ALttP Link can't become a wolf and have a way to fight back?