Ok, this is my reasoning. I am probably making some wrong assumptions here, pls tell me where I am going wrong.
Spin as a quantum phenomenon: Quantum phenomena disappear as the Planck constant goes to zero and since the intrinsic spins are proportional to the Planck constant, it is a quantum phenomenon.
Spin as a relativistic phenomenon: On other the hand, we know that the elementary particles are the irreducible representations(irreps) of the Poincaré group which is a purely relativistic symmetry group of the spacetime and since the spins show up in these irreducible representations. Therefore, the spins are a relativistic phenomenon. However, such phenomenon must disappear as the speed of light $c$ goes to infinity, but there is no $c$ in the spin value of the elementary particles,
so why do the spins appear in these irreps if they are purely quantum mechanical? can it be that they are both quantum and relativistic phenomena? if so, where is $c$?
After some thought, I've made some new observations. Wigner's classification of irreps of the Poincaré group doesn't specify the exact value of spin. It only tells us units (quanta) of spins (0,1/2,1,...). The exact values are determined by quantum mechanics. The Poincaré symmetry group(the symmetry of our universe spacetime) provides the boundary and the geometry. For a moment, let's consider the particle in a box, Schrödinger equation (quantum mechanics) gives us the exact values of the energy, but what determines the quanta is the shape of the box and the boundary (the symmetry). I am not sure if this reasoning is correct at all and I still don't get why wouldn't $c$ appear in the spin values.