First of all remember molecules and their solutions are oblivious of the labels and titles given to them by humans. It is our imperfect knowledge that we cannot fully describe them in words in a highly generalized way...the way mathematicians like to treat their ideas. To us mortals, space is the geometric space, say a 3D room, but for them space is an abstract set with certain properties (hard to visualize!). In the same way, acid and base are old terms and they meant different things to different scientists in their domains. It is a kind of a pigeon hole effect...same words being used to different observation/properties of molecules. Naunyn's work is almost exclusively in German in uncommon and now extinct journals. I could not find his original German paper of early 1900s that describes this definition but most of the sources are secondary. At least it is clear that he was a physiologist. To quote
Ole Siggaard-Andersen (1977) The Van Slyke Equation, Scandinavian Journal
of Clinical and Laboratory Investigation, 37:sup146, 15-20, DOI: 10.3109/00365517709098927 Link, subscription required to view
https://doi.org/10.3109/00365517709098927
A chemist in a glassworks might prefer a base to be a metal oxide. An organic chemist, or a biochemist, would possibly prefer to let acid
and base mean electron acceptor and electron donor respectively, but
we should then expect him to say Lewis acid and Lewis base. A
physiologist, or a clinician, might favour Van Slyke's description of
acid-base regulation and disorders, and would therefore be willing to
accept, e.g., sodium ion as a base, which, however, would not be
understood by an ordinary chemist.
So certainly sodium ion is not a base for a typical chemist but it is okay for a physiologist of his time. Sodium ion when paired with bicarbonate ion, forms a mildly alkaline solution. Similarly, sodium ion when paired with formate ion, the solution will be weakly alkaline. In this sense, sodium ion is a base to a physiologist...not to chemists.
Chloride ion is an acid in his mind because when it is paired with ammonium ion, it forms an weakly acidic solution.
The definition you cite, is thus specific to physiologists/blood biochemists. It is and was not a generalized idea. Call it a jargon rather!