I'm measuring genetic diversity from every individual in a sample using two different types of genetic information. These types of genetic information have different properties, and mutate independently of one another, yet will always be inherited together when an individual reproduces. Should I consider these different genetic measures as independent because they change independently? Or should I consider them dependent because they come from the same individual?
You should divide it into two parts.
Use the data and test for independence, ignoring the source to the data. Use the domain knowledge in order to think about reasons to the observed dependence or independence. The classical way to test that is the Chi-Square test of independence.
Both justifications you mentioned sound possible. Let the data decide which one is true.
You should note that if your data is small or biased, it may not represent the real relation. If you know of different properties in the data (e.e.g, other genetic information) that must be independent/independent, check them in your data set as a sanity check for being not representative.