Why is my wife shown as my 13th cousin?
Perhaps because she is your 13th cousin? You might use RelativeFinder to explore that relationship.
@John R Hager
That is always an interesting phenomena and similar relationships happen more that one would think.
Here is a link that will help you find the common ancestor between you and your wife.
In that relationship view you can scroll up and see how you are related and follow the two different lines. We are all one big family in many ways.
Hopefully this will help answer your question. Thank you for contacting FamilySearch. It has been a pleasure helping you. Best wishes as you continue finding records for your family.