I guess this is not the expected kind of answer, but you can study a metric space as a category, enriched over $\mathbb{R}_+$ - a monoidal category of non-negative real numbers (with $\infty$) with natural order and a single morphism $a\to b$ iff $a\geqslant b$. The monoidal structure on $\mathbb{R}_+$ is simply addition of real numbers $+$, considered as a symmetric enriched bifunctor. The original reference is "Metric spaces, generalized logic, and closed categories" by F.W. Lawvere. The category is associated in the most straightforward way: its objects are points of our metric space and $Hom(a,b)=dist(a,b)$. The law of morphism composition in a category is then precisely the triangle inequality. An enriched functor between such categories is a contracting mapping of metric spaces. Yoneda embedding embeds isometrically each metric space into a space of distance-decreasing functions equipped with sup-metric.

One should note that such generalized metric is not necessary symmetric and $>0$ for $a\ne b$, but in fact this isn't necessary to study it. One can always factor out zero-distance points and symmetrize the metric, and in fact this is a common procedure when one constructs new usual metric spaces.

A theory of Kan extensions becomes in this setting the following theorem: if $X \to Y$ is an isometric embedding, then any Lipschitz function on $X$ can be extended to $Y$ with the same Lipschitz constant. Among these there is the smallest and the largest one, corresponding to the left and right Kan extension.

Categorical limits become limits of sequences in metric space, and there is also a purely categorical formulation of Cauchy completeness condition. A category can be deduced from its category of $\mathbb{R}_+$-valued functors only up to equivalence - an equality of Cauchy completions. So complete metric spaces can be equivalently described by the metric spaces $\mathbb{R}^{X^{op}}_+$ of $\mathbb{R}_+$-valued contracting mappings on them. $\mathbb{R}_+$ has two natural monoidal structures: $+$ and $\times$. Both addition and multiplication of non-negative have a natural extension to bifunctors. They induce pointwisely two monoidal structures on $\mathbb{R}^{X^{op}}_+$, turning it into a semiring (no additive inverses). Thus I guess that you can take metric semiring of Lipschitz functions as an algebraic description of a metric space. There are purely categorical conditions for a category to be equivalent to some category of functors (smallness conditions). They can be used to tell if our metric semiring is really a semiring of Lipschitz functions on a metric space.

I guess this is not the expected kind of answer, but you can study a metric space as a category, enriched over $\mathbb{R}_+$ - a monoidal category of non-negative real numbers (with $\infty$) with natural order and a single morphism $a\to b$ iff $a\geqslant b$. The monoidal structure on $\mathbb{R}_+$ is simply addition of real numbers $+$, considered as a symmetric enriched bifunctor. The original reference is "Metric spaces, generalized logic, and closed categories" by F.W. Lawvere. The category is associated in the most straightforward way: its objects are points of our metric space and $Hom(a,b)=dist(a,b)$. The law of morphism composition in a category is then precisely the triangle inequality. An enriched functor between such categories is a contracting mapping of metric spaces. Yoneda embedding embeds isometrically each metric space into a space of distance-decreasing functions equipped with sup-metric.

One should note that such generalized metric is not necessary symmetric and $>0$ for $a\ne b$, but in fact this isn't necessary to study it. One can always factor out zero-distance points and symmetrize the metric, and in fact this is a common procedure when one constructs new usual metric spaces.

A theory of Kan extensions becomes in this setting the following theorem: if $X \to Y$ is an isometric embedding, then any Lipschitz function on $X$ can be extended to $Y$ with the same Lipschitz constant. Among these there is the smallest and the largest one, corresponding to the left and right Kan extension.

Categorical limits become limits of sequences in metric space, and there is also a purely categorical formulation of Cauchy completeness condition. A category can be deduced from its category of $\mathbb{R}_+$-valued functors only up to equivalence - an equality of Cauchy completions. So complete metric spaces can be equivalently described by the metric spaces $\mathbb{R}^{X^{op}}_+$ of $\mathbb{R}_+$-valued contracting mappings on them. $\mathbb{R}_+$ has two natural monoidal structures: $+$ and $\times$. Both addition and multiplication of non-negative have a natural extension to bifunctors. They induce pointwisely two monoidal structures on $\mathbb{R}^{X^{op}}_+$, turning it into a semiring (no additive inverses). Thus I guess that you can take metric semiring of distance-decreasing functions as an algebraic description of a metric space. There are purely categorical conditions for a category to be equivalent to some category of functors (smallness conditions). They can be used to tell if our metric semiring is really a semiring of distance-decreasing functions on a metric space.