Ising spin vs Pauli spin matrices
Neither, they are domain elements.  Any two domain elements would do: $\{+1, -1\}$, $\{0,1\}$, $\{\text{cat}, \text{dog}\}$.

Physicists have wonderful imaginations and thus name certain parameters based on their physical intuition for the behavior of nature.  In this case, they interpret the states of the particles interacting in the Ising model as either spinning up or down.  Furthermore, opposite spins produce opposite effects, such as their resulting magnetic fields being the negative of each other.  Thus, the state of the particles (i.e. the spins) are succinctly represented in the model as either $+1$ or $-1$.

Another good naming example from physics is in string theory.  The equations that describe the motion of these exceedingly small "objects" reminded the physicists of the equations that describe the motion of strings.  Thus the name.

The Potts model is a generalization of the Ising model.  One of the generalizations is that each particle can take one of $q$ values (i.e. the domain size is now $q$).  By continuing the intuitive naming from the Ising model, these $q$ states are also called spins.