I do not understand the confusion. The FDA clearly states that in order to be classified as a soap, the item must meet 3 very specific criteria. 1) it must be composed mainly of the alkali salts of a fatty acid, 2) the cleansing action must be caused only by those alkali salts of a fatty acid (no synthetic detergents allowed), and 3) its intended use. It must be labeled and marketed for use only as soap.
Shaving soap is not labeled as just soap, it is shaving soap.
Shaving soap is not used to simply clean your skin.
Shaving soap is labeled, marketed, and has a commonly accepted and intended usage for a purpose other than cleaning the skin.
Shaving soap is a cosmetic.
There have been a lot of discussions here and elsewhere about exactly how and why a shaving soap is NOT just a soap. You can't just take a good shower bar and rub some on your face and shave with it. Well, you could, but most shavers don't really want to. You really can't take a good soap recipe and throw a little clay in and have a great shaving soap. A shaving soap MUST be formulated a bit differently, despite being comprised mainly of the alkali salts of a fatty acid, in order to be good at doing what it is going to be used for; shaving. Not cleaning. If it was just soap then shaving soap would not need to exist. The fact that its intended use is for shaving makes it a cosmetic.
I would absolutely encourage anyone to research any question they have and get accurate information from the governing body rather than take the word of even well-meaning folks on an internet forum. It's great to share common interests and information, but that is no substitute for going to the source.
Back to the OP's question, yes you can say "made with skunk oil!" on the label, but the list of ingredients you posted would indeed be misleading. Although you are not required to list your ingredients on soap, you are required to follow the CPSC regs, and if you choose to list ingredients you must do it correctly.
But don't take my word for it