I'm not sure about the DNA sampling, but its been widely reported they did have large collections of "scent jars" which were just as powerful as having someone's DNA:

The Stasi had a whole range of methods and means to try to track down people who said or did anything critical of the East German communist regime. Collecting scent samples was used to try to identify those, for example, who had distributed flyers or who wrote critical graffiti.

When they found a piece of graffiti or a flyer then they took a dust cloth, which was usually yellow, and left it for a while lying next to the flyers covered by a protective piece of aluminum foil and then they had their sample. The cloth was then sealed in a pickling jar and stored. If the Stasi later came across a suspect in the process of the investigation, they tried to get a sample from this person as well -- of course, secretly. A trained dog was given the two smells, and if they matched, the Stasi had a concrete name.

https://www.dw.com/en/the-stasi-had-a-giant-smell-register-o...

Apparently back in 2007, German authorities were using this same method to track activists trying to disrupt the G8 Summit:

In a reminder of methods used by the East German Stasi secret police, German authorities are collecting human scents to trace activists they believe may try to violently disrupt the G8 summit in June. It's proving highly controversial, and there's no scientific evidence that the method is infallible.

https://www.spiegel.de/international/germany/stasi-methods-u...