Relations to group elements of a street together (associatedStreet is used more widely) are very rarely used by data consumers.
Transferring other tags to the relation would therefore effectively remove the road from the sight of renderers and routers.
In general it's just better to add your wikidata tag to each underlying way in the relation: it's not that hard to combine similarly tagged objects in post-processing OSM data. Routers and renderers do this all the time. General experience with associatedStreet has been that they can be quite hard to maintain, and are only worthwhile for use in unusual circumstances (i.e., it's better to place notionally redundant address tags than rely on a relation). I see no particular reason that type=street should be any different.
I think the list of relation types which work (i.e., are regularly consumed) is:
and no others, not even enforcement, although associatedStreet may be used in Nominatim.