The most general answer: It depends.
The crucial question is, what exactly do you mean by "voted similar". In statistics, often people tend to just use some given measure and then assume that it fits what they are testing. But it is important to first clarify what the terms mean.
As a somewhat sophisticated example. Assume Party 1 and Party 2 are two types of left-wing parties, and Party 3 is a conservative party. If you want to test if the political climate in the two states is similar, then surely differences between voter outcomes between Party 1 and Party 2 don't matter as much as differences in the outcomes between 3 and 1 or 2.
If you are just trying to find something descriptive, however, and don't care about deeper meaning of your concept, then the Pearson correlation can be a great tool, even when dealing with percentages (but beware the points below).
Most often, however, when people ask if such-and-such is correct then they are asking about inferences from these measures. So you might want to know if the p-value for a Pearson correlation on this data makes any sense.
Here are some key points, that might help you to decide this yourself (these apply to p-values from Pearson correlations):
- How many degrees of freedom does your data have (the data, not the correlation coefficient)? You might be tempted to say three, because there are three numbers. But are there really three independent measurements?
- Also, Pearson Correlations assumes normality. Can percentages ever be normal distributed?
Addendum:
Since there seems to be some confusion on why the Pearson Correlation might still be useful in this case, here a simple example.
Assume you clarify your question in the following way: "Given two random voters ($V_1$ and $V_2$), one from each state, what is the probability that both voted for the same party". This would give a simple measure of similarity, that also has a straight forward interpretation.
Let's do the calculation:
Assume the ratio of people who voted for each party in state 1 is given as $p_{1,1},\ldots,p_{N,1}$ (there are $N$ parties) and in state 2 it is given as $p_{1,2},\ldots,p_{N,2}$. The probability that $V_1$ voted for the first party is $p_{1,1}$ and likewise for all other parties. If $V_1$ voted first party, the probability that $V_2$ also did it (conditional probability) is $p_{1,2}$. Because those two voters can be assumed to be independent, the joint probability is $p_{1,1}\cdot p_{1,2}$. We have to sum up over all possible votes for $V_1$, so we get
$\rho = \sum_{i=1}^N p_{i,1}\cdot p_{i,2}$.
This is looking quite similar to a correlation coefficient, isn't it?