I notice that Valhalla and Graphhopper seem to treat bicycle roads as fine to route through for cars, see e.g. here: Linienstraße. OSRM, though, will route correctly.
In the example, it seems that Valhalla and Graphhopper both ignore at least three tags when routing for cars: bicycle=designated, bicycle_road=yes and vehicle=destination.
What can be done?
report that vehicle=destination is ignored for car routing on their issue tracker? Iff it is not yet reported and this tag is ignored. Note that vehicle=destination blocks only transit traffic.
You can also contribute code fixing the issue.
bicycle=designated does not mean that cars cannot go there, in the same as bicycle=designated on highway=path does not mean that pedestrian traffic is blocked.
I think, t is only the vehicle=destinateion that is being ignored.
In general, “Fahradstraßen” (bicycle roads) are allowed for other vehicles than bicycles as well.
I think they may be assuming that vehicle=destination is permitted if the endpoint is close enough, even if not precisely on the said road.
The restrictions for motorcars etc. are country-dependant: