From the point of view of the future owner:
Two of the bikes I've redone came to me with no identifying marks whatever and multiple applications of paint. I would have liked at least a headbadge, but didn't let their lack of pedigree keep me from building them into something unique.
One current project is a roadster bike, the frame of which is very generic. Stripping to bare frame provided no clues to identity, so I threw a wild black-with-antique-white-lugs paintjob on it. The frame didn't even have lugs, so it's a complete pretense. I'm thinking about making a headbadge for it out of Sculpey and epoxying it to the headtube as a goof.
If a bike is something special I suppose it is better to preserve its identity, but those kinds of bikes rarely fall into my hands...