Those are so called "Radschnellweg" (https://de.wikipedia.org/wiki/Radschnellweg), which literally means "bicycle highway". The signs that you posted seem to be part of the Radschnellverbindung Darmstadt-Frankfurt (https://www.nahmobil-hessen.de/unterstuetzung/planen-und-bauen/schneller-radfahren/radschnellverbindungen/radschnellweg-darmstadt-frankfurt/).
The green sign is the official StVO sign, i couldn’t find anything related to the black sign, but i would assume that this was used before the green one was officially added to the StVO, and is kept for some reason (maybe because some parts of the Radschnellverbindung are not yet up to the standard, where the authorities can add the official green sign, but still want to mark those parts to be part of the overall path). I could find that Hessen distinguish Radschnellverbindung and Raddirektverbindung, where the former has the official green sign and higher prerequisites than the latter one.
Credit:stackoverflow.com‘