How can a black hole not be a perfect sphere? As is stated or illustrated in many different articles, including this one, rotating black holes are oblate spheroids. It is likewise stated in some sites that this fact should be intuitive, as all rotating bodies are oblate spheroids. However, this is 1) not true, as objects which are perfectly inelastic will retain their original shape when spun, and 2) as far as I understand, planets and most other bodies gain this shape because of hydrostatic pressure, which obviously cannot apply to light and, again in my understanding, does not even exist in a black hole.
My question, then, is: why are rotating black holes not perfect spheres? Isn't gravity uniform in all directions? Or does the rotation of the black hole disrupt this uniformity?
Also, as a little bonus question: the first-ever direct image of a black hole, published today, appears to be oval-shaped. Is this because the black hole is rotating or some other effect?
 A: It is pretty difficult to answer "why" questions of this type. But to gain an intuition consider the following points


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*The rotating black hole is created by the collapse of a rotating cloud of gas or other matter. The particles on the overall axis of rotation carry no angular momentum with respect to the center, particles away from the axis do carry angular momentum on average. This means that the collapse is not spherical, the cloud is an oblate object at every step of the collapse. Why exactly should an object that is never spherically symmetric collapse to a single point? Instead, it first collapses to a disk-like structure with an oblate gravitational field. This geometry also survives in the curvature singularity of the black hole.

*The rotation of the black hole drags frames in its vicinity with a finite speed - that is basically how we came to the conclusion that the gravitational field represents a rotating black hole! However, the rotation speed has to disappear at some rotational axis and the gravitational field thus cannot be spherically symmetric.

*In a coordinate sense of the word, the horizon actually does appear at a single Boyer-Lindquist radius $r = r_H$. But the properties of the gravitational field at every point of this topological sphere are different when the black hole rotates (see previous points). So when we visualize or embed the surface $r=r_H$ in any reasonable way based on the actual physical properties of the gravitational field (space-time geometry), we end up showing it as an oblate surface as well.


As for your last note, the EHT image of a black hole shadow comes from a black hole immersed in a glowing plasma. The plasma is probably behind, next to, and even in front of the black hole. The shape of the silhouette you see cannot be quite understood as the shape of the black hole itself, but it is true that the spin plays a major role in the resulting image. 
Consider the following images, it is obvious from the sequence that an oblate shape of the shadow in the bottom right region is determined more by the geometry of the plasma than the oblateness of the BH field itself. The bottom right image actually corresponds, up to a rotation, quite closely to what EHT people believe is happening in the image of M87. (the image is from the simulations of Mościbrodzka et al. (2014)) 
