I make kombucha for another family member, who drinks it as a soda alternative. It doesn't fit in with what I define as appropriate foods for me. Kvass and beet kvass usually involve using grains or a culture such as whey, which I do not have or use. I wouldn't go as far as saying something was primal or non-primal for other people, but I do think it's a good idea to have a clear personal definition of what one considers appropriate.
I do eat fermented foods, such as a mild kimchi that I make at home. I only make kimchi from brassicae.