For context, Bundesnetzagentur (literally “federal network agency”) is a German agency that regulates five kinds of networks – electricity, gas, telecommunications, post, railways. The telecommunications part of their work is supervised by the Bundesministerium für Verkehr und digitale Infrastruktur (literally “federal ministry for transport and digital infrastructure” i.e. highways and data highways).

One of its tasks is to ensure non-discriminatory access to infrastructure markets for all market participants: Former state monopolies have to share their infrastructure at a reasonable price to competitors, for example. Another task is consumer protection: In february 2017 they decided that an internet-enabled child's doll containing a microphone accessible remotely without any indication to the people using that doll should be “verboten” since they classified it as a hidden wiretapping device.

The federal network agency claims that XMPP clients have to be registered according to a German law that says that one has to be registered with (but does not need permission from) the agency if providing public communication services commercially.

https://en.wikipedia.org/wiki/Federal_Network_Agency

Edit: As other hackernews have pointed out, only the telecommunications stuff is supervised by the BMVI.