Taking a look on Mapillary, traffic is regulated by a no-entry sign and not a designated cycleway as I had remembered.
It does look like it's oneway for buses, though, but certainly bidirectional for cycling (this also makes the footway tagging because that one certainly is a cycleway) but in any case, it's more than reason enough for me at least to tag the that segment of Grüneburgweg as highway=residential.

https://www.mapillary.com/app/?lat=50.12275930000001&lng=8.666541999999936&z=18.0245077791043&focus=photo&pKey=3846450802347722&x=0.5070865534758641&y=0.5062494807985585&zoom=0.0039676175570769745