I believe that per https://wiki.openstreetmap.org/wiki/Tag:highway%3Dcycleway
("When Not To Use") they should be mapped as part of the road.

The problem with doing it this way is that most bike tools such as PFB BNA assume that a separately tagged way is physically separated from the roadway and therefore lower stress. https://bna.peopleforbikes.org/#/places////

Similarly routing tools that optimize for low stress bike routes will get confused and present bad results to end users.

I'm in the middle of cleaning this up for you and retagging on the roadways!