tl;dr - It would be a huge coincidence if you found DNA that matched the actual perpetrator despite the fact that you collected it in the course of following the wrong guy, or canvassing for DNA at random.

Usually this happens by way of other non-DNA evidence. Imagine: you pick up a hair (or swab a coffee cup, whatever) in public that you believe (but, to your point, do not know) to be from your suspect. You take it back to the lab and, sure enough, it matches your sample from the crime scene.

Now, if you know nothing about the people in the vicinity of there you picked up your test hair, and were just randomly canvassing for DNA, this might not prove much. But presumably you were following a particular person, whose hair you tried to collect because you also have other evidence against him (though probably none so strong as DNA identification). This puts you in a very different epistemic situation with respect to that hair. Now you know that you picked up a hair in an environment full of people, only one of whom was a suspect in your investigation. It so happens that this hair matches a hair from the crime scene. It is, of course, possible that you picked up an unknown person's hair, and that person just happened to be your culprit -- but it is far far more likely that the hair you got is from your suspect, since he's the only one in the area believed to have any relationship at all to the crime.

(Of course, there could be situations that confound this analysis, for example if you collect hair at the end of the day from an interview room that's been used to question a bunch of suspects in the same crime. But law enforcement will not usually be this sloppy. They are well aware of the need to positively associate a hair with a particular person.)

One possible route: conduct the interview in a room that's been thoroughly cleaned beforehand (cleaners would need to be gowned). Take a sample from the chair pre-interview, then post-interview. Keep the room tightly controlled between cleaning and the interview.