From an SR point of view, I suspect that two things will happen. If the ball is moving across the observer's field of view, rather than directly towards or away, it will appear to be a prolate ellipsoid of revolution due to apparent length contraction. Additionally, the rotation will appear to gain a phase shift caused by differential time of flight of light emitted from different parts of the surface.
Rather more interesting are the GR implications, which I'm not qualified to specify. But I will point out that, at some scale the equator of the sphere will begin to look rather like a Tipler cylinder, aka a time machine. That assumes, of course, that Tipler is right and Hawking is wrong, and I'm generally reluctant to bet against Hawking. Since a "practical" Tipler cylinder only required neutronium and a tangential velocity of something like 0.5c, a sphere made of unobtanium ought to be serviceable.
This made me ask this question is taking the speed of light same in all directions an axiom of some sort?
Yes, although it is called a postulate rather than an axiom. This is Einstein's famous second postulate:
Any ray of light moves in the “stationary” system of co-ordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving body. Hence
$${\rm velocity}=\frac{{\rm light\ path}}{{\rm time\ interval}} $$
where time interval is to be taken in the sense of the definition in § 1.
A. Einstein, 1905, "On the Electrodynamics of Moving Bodies"
https://www.fourmilab.ch/etexts/einstein/specrel/www/
This postulate is simply assumed to be true and the consequences are explored in his paper. The subsequent verification of many of the rather strange consequences is then taken to be strong empirical support justifying the postulate. This is the heart of the scientific method.
So are all of our physics theories based on the assumption and what would happen if light turns out to be moving at different speeds in different direction? Will that enable transfer of information faster than the speed of light and is there any way for us knowing that the transfer happens faster than the speed of light?
Yes, all of our physics theories are based on this assumption, but the assumption itself is simply a convention. The nice thing about conventions is that there is no "wrong" or "right" convention. This specific convention is known as the Einstein synchronization convention, and it is what the second postulate above referred to by "time interval is to be taken in the sense of the definition in § 1". From the same paper in section 1:
Let a ray of light start at the “A time” $t_{\rm A}$from A towards B, let it at the “B time” $t_{\rm B}$ be reflected at B in the direction of A, and arrive again at A at the “A time” $t'_{\rm A}$.
In accordance with definition the two clocks synchronize if $$t_{\rm B}-t_{\rm A}=t'_{\rm A}-t_{\rm B}$$
A. Einstein, 1905, "On the Electrodynamics of Moving Bodies" https://www.fourmilab.ch/etexts/einstein/specrel/www/
If we define $\Delta t_A= t'_A-t_A$ then with a little rearranging this becomes $t_B=\frac{1}{2}(t_A+t'_A)=t_A+\frac{1}{2}\Delta t_A$. This is a convention about what it means to synchronize two clocks. But it is not the only possible convention. In fact, Reichenbach extensively studied an alternative convention where $t_B=t_A+ \epsilon \Delta t_A$ where $0 \le \epsilon \le 1$. Einstein's convention is recovered for $\epsilon = \frac{1}{2}$ and the Veritasium video seemed oddly excited about $\epsilon = 1$.
Note that the choice of Reichenbach's $\epsilon$ directly determines the one way speed of light, without changing the two way speed of light. For Einstein's convention the one way speed of light is isotropic and equal to the two way speed of light, and for any other value the one way speed of light is anisotropic but in a very specific way that is sometimes called "conspiratorial anisotropy". It is anisotropic, but in a way that does not affect any physical measurement. Instead this synchronization convention causes other things like anisotropic time dilation and even anisotropic stress-free torsion which conspire to hide the anisotropic one way speed of light from having any experimental effects.
This is important because it implies two things. First, there is no way to determine by experiment the true value, there simply is no true value, this is not a fact of nature but a description of our coordinate system's synchronization convention, nature doesn't care about it. Second, you are free to select any value of $\epsilon$ and no experiment will contradict you.
This means that $\epsilon=\frac{1}{2}$ is a convention, just like the charge on an electron being negative is a convention and just like the right-hand rule is a convention. No physical prediction would change if we changed any of those conventions. However, in the case of $\epsilon=\frac{1}{2}$ a lot of calculations and formulas become very messy if you use a different convention. Since there is no point in making things unnecessarily messy, it is a pretty strong convention.
Finally, regarding FTL information transfer. If we use $\epsilon \ne \frac{1}{2}$ then there is some direction where information can travel faster than $c$. However, since in that direction light also travels faster than $c$ the information still does not travel faster than light. It is important to remember that under the $\epsilon \ne \frac{1}{2}$ convention the quantity $c$ is no longer the one way speed of light, so faster than light and faster than $c$ are no longer equivalent.
Best Answer
In this unnatural setting (where classical electrodynamics exists and no cohesive forces with electromagnetic exchanges hold the ball together,) what will be happening as the equator of the ball approaches c and the central parts around the axis, the relativistic mass will be growing the further out from the axis .
Special relativity states for masses moving with a high velocity close to the velocity of light that the inertial mass changes , it is called relativistic mass, given by :
a body at rest has the rest mass
with the ratio
gamma goes to infinity as the velocity approaches c, and in the scenario above that is what will be happening to the density at the rim:
If the ball started with density d grams per centimeter it will become a function, d(r) where r is the distance from the axis. It will be the rotation of a non uniform ball . If a tangential force is applied to increase the outside equator motion, the mass there will grow accordingly to the impulse and a wobble will enter the uniform rotation.
In special relativity the extra energy entering a system when approaching the velocity c turns to mass , due to E=mc^2.
I will note that this is one of the few cases where the concept of relativistic mass is useful, i.e when one applies Newtonian physics where the second law holds with the relativistic mass.