If we consider a symbolic description of a Kripke structure (I guess that is what you mean), then we have the variables V, the initial states I, and the transition relation R given. The states are by definition the subsets of V, i.e., we have 2^|V| many states and these subsets are viewed as variable assignments to find the transitions with R.

So, there is no label function in this case, because we define L(s) = s and that in turn means that there is no reason to have your doubt.

The other way around: if we have a Kripke structure with two states s1 and s2 with the same label L(s1)=L(s2), then we cannot generate a symbolic description just with the variables known by the structure. In that case, we have to add further variables to distinguish between such states. And such additional variables may have to be ignored for simulation computations etc. That is one reason why these different sets of variables may come from.