Is there a word for a last name that is also a word? If so, what is it?
Solution 1:
I think the confusion you are facing is as a result of the fact that we have last names in English that are originally from other languages, where, in those languages, these last names are also meaningful words. To use your example, we have such last names as Braun, (German for "brown"), etc, which don't actually mean anything in English.
It is facing the contrast of such last names with English last names, which are meaningful words, that make them stand out.
Many last names come from professions (Taylor), and others from the compound of lineage (Johnson = "son of John"), so in reality, to some extent or another, all last names are "words" in the sense you are looking for, because all last names mean something... even if it's in another language, or their spelling has changed slightly over the years (as "Taylor" has).
So perhaps the answer to your question is a word hat means: "meaningful word in English" or "word which means something in English, and perhaps the best single-word that (unequivocally) means that is "sensical". There is also "meaningful" (as mentioned before), "coherent", "intelligible", "significant" and their synonyms, but these also have other, stronger connotations that might obscure what you mean.