I think the untranslatable self refers to the idea of not being able to put a person’s identity into words, into categories. We are who we are. We are not just a gender, or a social class, or a race. We are human beings. We can use words to communicate. We can use words to disagree or praise someone.
We can translate something from Spanish to English, from French to German, but sometimes there are things so unique to a language that they are untranslatable quite literally. So, to me it seems that Whitman finds people to have that same uniqueness that is untranslatable. We can’t put a soul into words.