@Yemon. I didn't highlight that metrics have not to be symmetric because it seems me something implicit in the request. But ok! Now I see that it wasn't clear, and I could edit the OP and stress that "metrics" are not necessarily required to have all the features of standard metrics and norms are not required to have all the features of standard norms, though I don't really know what *you* would identify as the "features of standard norms": Normed groups, valuated domains, vector spaces, C^*-algebras are equally interesting examples of algebraic normed structures, but they have dramatically different characteristics. E.g., suvrit pointed out that vector space norms are convex, but this does not make sense in normed groups. And you observed that *common* normed structured come with a distinguished element whose norm is zero, but this is not the case with normed semigroups. So, at the end of the day, I don't understand how this kind of indirect arguments should prove anything in relation to my question. Apropos, let me repeat it once more: Does there exist any previous attempt to construct a unified framework where metric spaces AND algebraic normed structures can be all regarded as [special] instances of the one same archetypical structure? The term "structure" is formally defined in the OP. Now, what is different from asking: Is there a unified setting to carry out the study, say, of continuity, compactness and connectedness? Would you have considered that as a vague question too? I don't think so. Ok, you would have considered it as a trivial question, but that's not the same. P.S.: In any event, I've just voted up to close the thread.

@Andreas. Yes, surely! The framework should not be too general if the goal is to make it *useful*. Here, *useful* means, for instance, that typical theorems or constructions commonly encountered in this or that particular context - and I have listed normed groups, valuated rings, etc - can be proved or accomplished once and for all, without any need to replicate them in every each case as if they were totally independent from each other (that's why I made explicit reference to products and coproducts in the OP). From a more philosophical outlook, a "good unifying framework" should suggest new directions of research and provide insights for a better understanding of the ontology of metrics and norms (in the same spirit of Lawvere's work on metrics and metric spaces), e.g., by smoothing out, from the natural perspective of categories, a certain degree of arbitrariness which *happens* to be inherent, at least to my eyes, to the classical approach to the foundations of the theory of metric spaces (I've already alluded to this in reply to a comment by Pietro Majer), to the extent of separating, as I tried to say somewhere else, what is essential from what is accidental (in our standard vision).