Why should $T : D∗ ⊂ R ^2 → D ⊂ R ^2$ generally be injective and surjective if we want $T$ to represent a choice of coordinates for the region $D$?
My thinking is that if it's bijective then it'll map exactly to the same spot on the transformation because it's onto and one-to-one. But I feel there's a little more missing and I'm not wording it 100% accurately.
It has nothing to do with "same spot".
You always want each point to have coordinates. In your original coordinate system each point had coordinates. You want your transformation to be onto so that each point still has coordinates in your new coordinate system.
You always want every pair of different points to have different coordinates. In your original coordinate system every pair of different points has different coordinates. You want your transformation to be one-to-one so that every pair of different points in the new coordinate system also has different coordinates.