Finding similarity between two datasets
I would say that they are probability distributions. You can interpret them as the probability that a randomly drawn person from a given county belongs to the white majority. However, they are not probability vectors, as this would require them to sum up to 1. It's more that each element of the array defines a distribution, e.g a random person from county 1 is white with a probability of 0.52 and a member of a minority with probability 1-0.52=0.48. If you want to compare this distribution with your simulation, you would have to make the comparison element-wise, e.g. compare the 0.52/0.48 actual distribution with the 0.48/0.52 simulated distribution. This is also why reshuffling leads to different results: suddenly you are comparing the actual distribution from county 1 with e.g. the simulated distribution from county 5. This does not make much sense. So if you shuffle, you need to shuffle both arrays in the same way.

As you can interpret the distributions in each of the counties as being independen from each other, you can compute DKL or cosine or Earth's Movers for each county and sum them up probably.

I'm not sure which method is the best. I would probably go with DKL out of habit. You could compute it as such

dkl = 0.52*log(0.52/0.48) + (1-0.52)*log((1-0.52)/(1-0.48)) + ...


I hope this helps!