There are several results in topology about metrics taking values in (the positive cone of) a partially ordered Abelian group $\mathbb{G}=(G, <, +, 0)$. The first occurrence of this spaces I'm aware of is dated 1950 by Sikorski (MR0040643 - R. Sikorski, Remarks on some topological spaces of high power). He literally started a new branch of topology that investigated this kind of objects, called $\omega_\mu$-metrizability.
There are also several results about the topological nature of spaces with metrics taking values in other kind of structures. The wider framework I'm aware of has been studied by Reichel in 1977 (MR0458373 - H. C. Reichel, Some results on distance functions). I'll resume here some of its results:
He consider totally ordered semigroups with minimum $(S,<, 0, +)$, i.e. structures such that $(S,<, 0)$ is a total order with minimum $0$, $(S, +)$ is an Abelian semigroup and $<$ is translation invariant, i.e. $a<b$ implies $a+c<b+c$ for every $c$ (I believe you also wanted to assume this kind of invariance?).
We say that a $S$ is continuous if whenever a sequence $\langle x_i\mid i<\gamma\rangle$ is coinitial in $(S\setminus\{0\},<)$, then $\langle x_i+x_i\mid i<\gamma\rangle$ is coinitial as well in $(S\setminus\{0\},<)$. Notice that every structure of the form you described is a continuous totally ordered semigroup with minimum in this sense.
Let us call $S$-metrizable a space that has a metric in the structure $S$. It turns out that the behavior of such class of spaces is influenced mostly by the coinitiality of $(S, <)$, i.e. the size of the smallest sequence of elements of $S\setminus \{0\}$ converging to $0$. We call such cardinal number the degree of $S$.
Reichel proved the following:
Theorem 2, countable (Reichel): For a given topological space $(X,\tau)$, the following are equivalent:
- $X$ is $S$-metrizable for some continuous totally ordered semigroup $S$ of degree $\omega$.
- $X$ is metrizable (with standard metric over real numbers).
(He also characterized spaces that are metrizable over non-continuous totally ordered semigroup $S$ (Theorem 1 of the paper cited above)).
When the degree of $S$ is uncountable, than situation is easier.
Theorem 2, uncountable (Reichel): For a given topological space $(X,\tau)$, the following are equivalent:
- $X$ is $S$-metrizable for some continuous totally ordered semigroup $S$ of degree $\kappa>\omega$.
- $X$ is $S$-metrizable for $S$ the set of positive elements of some totally ordered Abelian group of degree $\kappa>\omega$.
Combining this with other results in literature, in second case one get the following:
Theorem: For a given topological space $(X,\tau)$, the following are equivalent:
- $X$ is $S$-metrizable for some continuous totally ordered semigroup $S$ of degree $\kappa>\omega$.
- $X$ is $S$-metrizable for every continuous totally ordered semigroup $S$ of degree $\kappa>\omega$.
- $X$ is $S$-metrizable for $S$ the set of positive elements of some totally ordered Abelian group of degree $\kappa>\omega$.
- $X$ is ultrametrizable over some total order of coinitiality $\kappa>\omega$ (i.e. the $+$ operation of $S$ coincide with the maximum between elements).
- $X$ is a subset of $\lambda^\kappa$ with bounded topology for some $\lambda$.
- ...(there are several other topological characterizations).
If you want to know more about this, L. Motto Ros, P. Schlicht and I recently wrote a paper on the subject (available on arxiv at this link). In that paper there are also several other references to the literature.