That's ridiculous hyperbole. The stasi didn't spy on internet communications, while lots of governments in liberal democracies do. So the only thing you have to link the Stasi to the NSA is the fact that they are intelligence organizations.
The Internet wasn't a thing. The link isn't in terms of the specific things they do, it's the attitude. The Stasi were known for wanting to know everything about everyone and using questionable tactics to do so. That is exactly what the NSA seems to be moving toward.
