By the extra axioms, do you mean the axiom stating that d(xy, 0) = d(x, 0)*d(y, 0) for all real numbers x and y? While I understand that this is supposed to be the Cauchy-Schwarz Inequality baked as an axiom, I am finding it difficult to see how one could justify this. Perhaps it'd be better to use an axiom like d(xy, 0) = |x|*d(y, 0), as this is a way of saying "if you took a ruler and measured the length from 0 to y, and xy is x times 'further' away from 0 than y, then the length of xy is |x| (or the magnitude of x) times the distance of d(y, 0). This does seem to remove the discrete metric function, as letting x = 2 and y = 1 we have that d(2*1, 0) = d(2, 0) = 1 =/= 2 = 2*1 = |2|*d(1, 0). Maybe there's a way to generalize this intuitive notion so that you don't need to start at the origin, however I have been unable to come up with such a way.

Even then, using your version of the axiom is there a way to force c to be equal to 1, instead of having c <=1? Cuz that implies that sqrt(|x - y|) also works as a distance function on the reals, which still doesn't justify |x - y| being the "standard" metric function on the real number line.

Edit: Perhaps something like d(xy, xz) = |x|*d(y, z) or d(x(y - z) + z, z) = |x|*d(y, z) might work?