Physical significance of the discrete Heisenberg group

In quantum mechanics, Schrodinger’s position and momentum operators are characterized by the fact that they form a representation of the Heisenberg group $H_{2d+1}$ over $mathbb{R}^{2d+1}$.

Discretizing this group, as did Weyl, gives a Heisenberg group over $mathbb{F}_p$ for all prime $p$, as described here.
In the case $p=2$ this is the group generated by the Pauli matrices $sigma_X = begin{pmatrix} 0 & 1 \ 1 & 0 end{pmatrix}$ and $sigma_Z = begin{pmatrix} 1 & 0 \ 0 & -1 end{pmatrix}$.

The same Pauli matrices happen to generate the algebra of observables for a spin-$frac{1}{2}$ particle.

Is this a pure mathematical coincidence? (If it is not a coincidence, then how does one mathematically distinguish the two situations?) Or is there a physical explanation for this observation that "discretizing the algebra of observables generated by position and momentum in continuous space gives the algebra of observables for a spin-$frac{1}{2}$ particle"?