How do I center an object in its parent? I have 2 objects.
Their dimensions are completely identical.
I have applied all transformations, rotations and scales.
Their locations are 0, 0, 0.
I have tried parenting the objects, both in relations and object constraints. Neither parenting nor making them separate objects has any effect.
Fiddling with their origins has no effect.
And yet, I can never get the objects in the same position. They are always off:

How in the world is one supposed to center two objects into each other? It seems that no matter what I do, they are never in the center of each other, despite having exactly the same dimensions.
As you can see the problem still occurs in-game:
Your problem is essentially this one. With your current Pivot Point settings, ('Median',) the center of your object is considered to be the mean location of its vertices, not the center of its bounding box.

Set Pivot to 'Bounding Box'
Edit Mode, all selected, ShiftS,Cursor to Selected
Object mode, ShiftAltX Origin to Cursor
.. and set the location to 0..

...for both objects.