Yep, it’s because they are referring to the town (Baden) in the state (Baden).
A few extra bits and pieces.
First of all, it is a stunningly gorgeus, rhapsodically beautiful place to visit. If you get the chance, GO.
Also, the Wiki quote above doesn’t quite bring out the ‘cause and effect’ involved. The French decided early on in the war that after the war BB would make quite a nice new base, and for this reason they decided never to attack it or bomb it. Which is why it is so beautifully untouched and preserved.
Thirdly, you can get quite a fun trivia quiz question out of the town, eg ‘In which European town are all the drivers bad?’ or something like that. This is because all the local licence plates all begin with ‘BAD’. It’s true (I even have a photo somewhere).
If you want to see some pix of this truly beautiful place, I’ve got some here.