I don’t think that keys/values from referenced rhino geometry are passed/stored in GH. (Maybe something has changed in recent versions?) As far as I know, that info is stored in the rhino document. But, I guess you could store the GUID of the rhino object somehow, and then use that to read from the active rhino doc?
Here are some older examples of how to retrieve user text, (keys/values), from Rhino Objects.