Does this unexplained phenomenon have direct implications to the the
Big Bang theory
Not necessarily, although it depends to some extent on how you define the contours of Big Bang theory.
It does not (at least as a matter of necessity) contradict the notion that all of the mass-energy of the universe was confined to a very small space about 13.7 billion years ago that rapidly expanded, that the universe continues to expand, and that the decreased density and temperature of the universe caused it to evolve in particular ways (like Big Bang Nucleosynthesis) that gave rise to our current universe. This does not inherently require spherical symmetry, homogeneity, or a lack of anisotropy.
Explosions that are disk-like rather than spherical are common place in the ordinary world of classical physics and there is nothing inherent in the Big Bang Theory that precludes a non-spherically symmetric "bang" in that context.
and Cosmological inflation and if yes which are these?
The "axis of evil" if it is more than a systemic or methodological artifact of how we can gather astronomy data from our current location in the universe is evidence that the universe is anisotropic and not homogeneous.
One of the main purposes of cosmological inflation as an explanatory theory is to explain why the universe is isotropic and homogeneous to a greater extent than one might expect in the absence of cosmological inflation:
Many physicists also believe that inflation explains why the universe
appears to be the same in all directions (isotropic), why the cosmic
microwave background radiation is distributed evenly, why the universe
is flat, and why no magnetic monopoles have been observed.
While it doesn't necessarily rule out any and all possible cosmological inflation theories, almost all of them assume (in part for ease of analysis but in part for other sound reasons) spherical symmetry as a core raison d'etre.
But, if the axis of evil is a real phenomena, then it almost necessarily follows that this structure was seeded prior to or during cosmological inflation. This is because cosmological inflation is used to explain the large scale structure of the universe and the "axis of evil" if real is the largest scale observed feature of the large scale structure of the universe. So, if the axis of evil is real this implies that a very different mechanism, perhaps a variant of cosmological inflation or perhaps something entirely different, explains the qualitative features of the universe we observe which cosmologists frequently assume are due to one of the hundreds of competing cosmological inflation theories that has been proposed so far.
Of course, it is also worth noting that while the Big Bang theory in some permutation or another has almost universal acceptance among astrophysicists, cosmological inflation does not have consensus support, let alone consensus support around any particular mechanism of that theory. Even one of the original inventor/discoverers of the concept, Alan Guth, now seriously doubts it.
An alternative explanation is that cosmological inflation which in the most common versions of this class of theories cannot explain this observation, is wrong and that the features of the large scale structure of the universe which cosmological inflation seeks to describe have a different source.
It isn't fruitful to speculate too deeply on what alternatives to cosmological inflation could involve, even though there are a handful of proposals out there, such as the torsion-based cosmological model proposed by Nikodem Poplawski mentioned by @Edouard, since none of them have wide acceptance or have been seriously vetted by others in the field. If the "axis of evil" observations start to look like they are real and not just methodological artifacts, then in due time new theories will develop to explain it whose features are hard to predict if one is not omnipotent.
nobody seems to address (besides the Copernican Principle violation)
the "elephant in the room"
These seem like two sides of the same coin to me, rather than distinct concerns. Cosmological inflation and the Copernican Principle are deeply intertwined. The former is a common and standard explanation for the latter (although not the only one).