Strange as it may seem, to me all unscented soap has a scent, but they don't all smell the same, either. I know the general discussion here is probably thinking about people with sensitivities or allergies to certain ingredients in fragrance oils or essential oils, but there are instances where the smell of some plain soaps with no added scent may be an issue as well. (For example, take a neem oil soap to which no fragrance is added. Or DeeAnna's pine tar soap example. And the oft-discussed lard soaps. Or in some cases pure olive oil soaps - some people can't stand the smell of olive oil soap.)
I think that for the person who is sensitive to odors (not necessarily FO's themselves) even plain unscented soaps might smell as if it is scented. Last year I gave some unscented soap to a lady who insisted that it had lavender fragrance added, when I knew it positively did not and had never even been stored near a lavender fragranced soap ever. But of course, I also have a family member who says, 'I don't do unscented' who won't even look at, touch or sniff a soap that I say it is unscented.
But to the original question, 'aren't all fragranced soaps considered cosmetics?' I wonder where the question is coming from. In re-reading the question, I suppose the OP is referring to 'intent' of the soapmaker. I suppose if the soapmaker's
intent is to fragrance the body, then yes, I suppose it would be, but only if so labeled. But I don't think that is the intent of most experienced soapmakers because we know it is a rare soap that actually leaves a long-lasting fragrance on the skin. And even then, when it does happen, it doesn't happen to everyone, as mentioned in another thread, where the fragrance in the same soap left scent on the husband's skin, but not on the wife's (I'd link it, but can't find it now) and/or not everyone has a sensitive enough nose to smell it lingering on the skin. (Often my husband who smoked for decades can't smell things I can smell. Is my nose more sensitive because I am not a long-term smoker? Or is it just the way my nose is regardless? I don't really know, but I do know smoking does interfere with the sense of smell as do some drugs and other factors can as well.)
So, No, soap is not a cosmetic by the US FDA's standards simply based on added fragrance. Published claims about the soap is more to the point when the FDA determines how it should be categorized. Although there are cases where consumer perception might be included in the FDA's determination of whether a particular soap is considered a cosmetic (or a drug).
https://www.fda.gov/cosmetics/guidanceregulation/lawsregulations/ucm074201.htm