Suppose there is a rod, on the wall or at the centre, in a pressurized vessel, the pressure inside the vessel being P. The rod has a shear strength of S. Is it true that the rod will break, in shear, if P>S, otherwise it will not? To me it seems so, but I am not able to get my head around it.
It won't. Solid objects don't break in hydrostatic compression. The pressure is pushing all parts of the rod together, so there's nowhere for them to go except stay together. You can imagine an object on the ocean floor. Pressure could easily exceed shear strength but even soft animals remain intact, as do rocks.
The von Mises failure criterion says a material that follows it can sustain infinite hydrostatic pressure or tension without reaching the failure surface. See the diagram here. The yield surface is a tube that extends out to infinity in the hydrostatic compression direction.
There can be other failure modes, like phase change. Or it could collapse if it contains air pockets. But that's not determined by the pressure exceeding the shear strength.