I do not think it would be easy but with advances in testing and the budget of a 3 letter agency I have no doubt it could be figured out. The way I saw it happening in my mind was somewhere after the flush down the pipe a sample would be extracted.

This is more in the realm of conspiracy theory but I have wondered about it once upon a time thinking of book ideas. But the reality is that this idea is way to complicated. Instead many people willingly have their DNA tested and information is put into a database which I am sure every 3 letter agency already has full access to. Also do a crime and you and your future and past family DNA privacy is gone after a court ordered sample exposes you and all your family. We also have situations like cord blood being tested after birth without permission. So in the end no complicated system is really needed. I think DNA privacy is doomed in the end. Which in some situations is good (horrible crimes) but in others will be really bad (denied insurance based on genetics).