The Bugatti marque (original) died somewhere like 1951, their glory days were definitely the 20's (the Type 35 still holds records for number of races won) and 30's. There was a shot at bringing the marque back in the 1990's, short lived, although they turned out an interesting car or two. Then Volkswagen got the rights to the name, and that's the current Bugatti.

Ettoir Bugatti did build a couple of bicycles back in the 1930's. They're insane looking, aluminum if I remember correctly, and go for insane prices nowdays - assuming you'll find somebody stupid enough to want to sell it in the first place.

Your bike is just using the name - and there's a pretty good chance it was used illegally at the time. Sort of like Indian motorcycles: After the 1950's, the legal trademark got rather murky with all sorts of people playing fast and loose with it. Usually attempts to bring back the automobile, which never came to anything.