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For space reasons I'll cut my reply into two parts. The first discusses the Copernican principle in question, the second answers your question about this specific blog post.

The modern understanding (and name) of the Copernican principle is really owed to mid-20th-century Hermann Bondi's work in general relativity, and it is a generalization of the initial Copernican model of heliocentricity, with the sun at the centre of the universe, and the Earth, other planets, and distant stars tracing out exactly circular concentric orbits around it.

At its most general while still retaining its strength, the Copernican principle says that in a system with certain symmetries, there is no distinguished position on a circular orbit. This is not just orbits within a 4-dimensional spacetime; it applies in certain many-dimensional phase spaces too.

(One can generalize further by giving up some strength and say that most spaces with certain symmetries admit a notion of typicality which applies to any choice of initial momentum almost everywhere in the space. We can then discuss how such a measure breaks in more complicated systems. Consider translational symmetry on the Earth on an overcast moonless night. If you choose a random spot on Earth and then swim or walk a kilometre or ten in any direction, your view of the surface features out to the horizon is unlikely to change. If you found yourself somewhere in a salty body of water nowhere near land you would struggle to tell with any precision where on Earth you were or which compass direction you had moved. If you found yourself somewhere in a sandy desert or flat scrubland far from human settlement and no "celestial guides" like the position of the sun, again you would struggle to tell with any precision where on Earth you were or the direction you are facing. There are however atypical features of the surface of the Earth which break translation symmetry: coasts, edges of forests, peaks of mountains, human settlements, Manhattan, you name it. Moving from water to land or vice-versa clearly breaks some global notions of typicality. However there is a lot of coast on the Earth. You'd probably only find complete atypicality when close enough to major landmarks like the Great Pyramids of Giza or Niagara Falls. We can also add in a notion of temporal typicality -- sufficiently close to sunrise or sunset, or on starry nights, it is easier to orient oneself towards compass points.)

Our solar system's mass distribution is only approximately spherically symmetric, and planetary orbits are non-circular ellipses, so Copernican heliocentricity holds only approximately. And of course we now know that other stars do not orbit our own (even nearby ones do not move in a circular or even elliptical orbit around it). The Copernican approximation is still locally useful as a basis for comparison with observations, and those led quickly to Kepler discovering the features of stable elliptical orbits, Galileo discovering the large moons of Jupiter and their orbits around it, he and others the phases of various planetary bodies, and ultimately Newtonian gravitation.

Copernican heliocentrism is thus correct in some effective limit: it works as long as we do not look too closely at small details of the sun's wobbles or perturbation of various orbits by Jupiter, and as long as we are only considering things at a solar system scale. (It applies in many other solar systems too: a central mass tends to entrain smaller masses into nearly-circular orbits. And it is useful for comparison studies of star systems where orbits are far from circular (many many comets, strange exoplanets) or where there are two or more stellar masses surrounded by smaller bodies.). And that it is not exactly correct made (and still makes) it even more useful in exploration of the real solar system.

There is a notion of Copernican typicality in galaxies too. There is nothing clearly special about our solar system's place on its orbit through the Milky Way, thanks to the galaxy's approximate axisymmetry. Likewise, except close to the galactic centre, the galactic edge, or well outside the plane of the disc, virtually all star system orbits through the Milky way are highly typical. As in Copernican heliocentrism where the position of Earth at any time isn't particularly special, "Copernican Sgr A* centrism" means the position of the sun isn't particularly special either. Of course we are adapted to "goldilocks" atypicality at the solar system scale: we thrive in a family of approximately 1 a.u. orbits, and would struggle to survive in most others. We are less sensitive to the path our solar system takes through the galaxy.

Next, there is the Copernican principle in cosmology. This Cosmological Principle starts with the greatest symmetry: a universe which looks the same in every direction from every possible vantage point. Cosmology is in many ways a study of how the Cosmological Principle breaks down. It does in various ways:

* Locally, we're on a planet. On a starry night, down looks very different from up. There are our solar system's planets in various directions but not in most directions.

* We're also in a galaxy in a local cluster: from the southern hemisphere we see the galactic bulge. Dust and gas distributions are denser in some directions than others. We can also see satellite galaxies like the large Magellanic cloud. We can also see M31 taking up a surprisingly large solid angle of the sky (more than the moon; it's just that the Andromeda galaxy is dim), and a handful of others.

* There is a Cosmic Web structure to bright clusters of galaxies, and (misleadingly named but definitely sparser) "supervoids" between the filaments of the web. There are also smaller overdensities and underdensities in these large regions.

However, we have not done much damage to the typicality of our vantage point. There are lots of similar star systems in lots of similar galaxies in lots of similar clusters in some area of similar Cosmic Web density.

It is clear, though, that the entire cosmos is not uniform, that there are various boundaries against which one can take an orientation.

Cosmology is also about understanding temporal typicality too. Even in the very early 20th century, there were questions being studied: has the universe always looked like this, with lots of galaxies containing many stars like our own? Trying to answer this question with the hard requirement that any answer be in concordance with available evidence led us to a definite no. In the reddest, dimmest, smallest-solid-angle galaxies we have observed, our sun would be extremely atypical based on its spectral lines indicating elements other than hydrogen and helium, but in less-red, less-dim, less-small-solid-angle galaxies (right up to M31) we see that our sun would be less and less unusual.

We can then think of temporal boundaries: what happens in the very earliest times? What's at the highest redshift (answer: the cosmic microwave background and no bright galaxies)? What does that imply about the deeper past? Does it imply anything about the future?

The concordance cosmology -- the standard cosmology, with \Lambda-CDM serving as its mathematical expression -- works exactly with all these various "distractions" smoothed out. Rather than considering star-filled galaxies forming clusters of various sizes, we consider a dust uniformly scattered through an expanding space, with non-gravitational interactions (radiation pressure) becoming more important in the past and less important in the future.
This picture is then deliberately perturbed with features found by astronomers, and those perturbations are studied for their impact. Most remain local, a scale far far from cosmological.

Informally, this means "We don't care what happens inside individual galaxies, each of which is just one mote of the cosmological dust, and we track the components generating the energy-density of a typical point in space as the dust -- or its various components -- dilute with the expansion, or grow denser as we look into the deep past".

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