I don't do SNP design but I don't think 23andMe made their own chips at first (if they have ever). SNPs are chosen based on being informative for population discrimination, with traits being overlaid afterwards
I could be wrong, but I do believe they get to choose. What I do know is that none of the big testing companies test exactly the same SNPs, there is some overlap but every test is different. They also change their SNP sets from time to time, 23andMe has changed at least 5 times, and interestingly, their current test isn't the biggest (it tests fewer SNPs than some of their earlier ones).