If the features are not statistically independent, I don't think it's likely that they're truly orthogonal; which might not affect the utility of a distance metric that assumes that they are all orthogonal.

You're exactly right, this whole thing is indeed a bit of an obvious nothingburger.

Most of calculus and undergraduate math, engineering, and physics takes place in Euclidean space R^n. So all the curves and surfaces directly embed into R^n, usually where n = 2 or n = 3. However, there are more abstract spaces that one would like to study and those are manifolds. To do calculus on them, they need to be smooth manifolds. A smooth structure is a collection of "patches" (normally called charts) such that each patch (chart) is homeomorphic (topologically equivalent) to an open set in R^n. Such a manifold is called an n-dimensional manifold. The smoothness criterion is a technicality such that the coordinates and transformation coordinates are smooth, i.e., infinitely differentiable. Smooth manifolds is basically the extension of calculus to more general and abstract dimensions.

For example, a circle is a 1-dimensional manifold since it locally looks like a line segment. A sphere (the shell of the sphere) is a 2-dimensional manifold because it locally looks like an open subset of R^2, i.e., it locally looks like a two dimensional plane. Take Earth for example. Locally, a Euclidean x-y coordinate system works well.

Euclidean space is a vector space and therefore pretty easy to work with in computations (especially calculus) compared to something like the surface of a sphere, but the sphere doesn't simply abandon Euclidean vector structure. We can take halves of the sphere and "flatten them out," so instead of working with the sphere we can work with two planes, keeping in mind that the flattening functions define the boundary of those planes we're allowed to work within. Then we can do computations on the plane and "unflatten" them to get the result of those computations on the sphere.

Manifolds are a generalization of this idea: you have a complicated topological structure S, but also some open subsets of S, S_i, which partition S, and smooth, invertible functions f_i: S_i -> R^n that tell you how to treat elements of S locally as if they were vectors in Euclidean space (and since the functions are invertible, it tells you how to map the vectors back to S, which is what you want).

The manifold is a pair, the space S and the smooth functions f_i. The smoothness is important because ultimately we are interested in doing calculus on S, so if the mapping functions have "sharp edges" then we're introducing sharp edges into S that are entirely a result of the mapping and not S's own geometry.

Sagan's video proves that it's always helpful to investigate abstract concepts with an experiment.

But the video also raises an important question: Can we derive true conclusions from wrong assumptions?

Here the assumption that the "flatlanders" are really flat is wrong.

Sagan notices that too and he says that his objects are not really flat, but they have thickness.

  Now these little cutouts have some little height 
  but let’s ignore that, let’s imagine that these 
  are absolutely flat.

This is the typical rhetorical sophistry widely used in physics. We may also call it equivocation on the word zero. Here, Sagan defined the word zero both as nothing and something and he wants us to go along with this equivocation.

You cannot have non-zero height and zero height at the same time. If you have zero height you do not exist. A door is either open or closed.

Instead of flatlanders, Sagan may have used shadowlanders. Shadow is closer to two dimensional objects but even shadow has thickness.

So, Sagan's assumption that his objects have no height is definitely wrong. Can we arrive at a correct conclusion from this wrong assumption? I guess not.

Also it is clear that when we try to transform a 3-D object into a 4-D object, the object's morphology changes. A cube becomes something else. A cube does not cross the dimensions as a cube.

This also raises questions about physicists' claim that we are living in a 4 dimensional continuum called spacetime. If so, our current morphology in the 3-D world and 4-D world of spacetime cannot be the same.

Then the question is, is our present anatomy 3-D anatomy or 4-D anatomy?

Are you sure? In a 3-D coordinate system all axes are orthogonal because they make 90 degrees angles with each other.

Edit: I see that you mean "cannot be drawn as orthogonal lines on paper." But, in reality they are orthogonal e.g., when I construct a 3-D model of a cube.