What's $\mathbb{R}$ doing in the definition of a metric space?

Flagg in "Quantales and continuity spaces" defines the notion of a value quantale, a certain complete lattice satisfying a short list of very reasonable properties. He then goes on to define a continuity space to be precisely what you are suggesting: a metric space valued in a value quantale. The classic notion is recovered since $[0, \infty ]$ is a value quantale. There are many other value quantales giving rise to truly different kinds of 'metric spaces'.

Flagg's work builds on previous work of Kopperman where he essentially did the same thing but used a value semigroup instead of a value quantale. There are some distinct differences between the two approaches. For instance, defining point-to-set distance in the usual (obvious) way, in Flagg's formalism the point-to-set distance from $x$ to $C$ is $0$ if, and only if, $x$ belongs to the usual topological closure of $C$. This property does not hold in Kopperman's formalism. Also, in both formalisms every topological space becomes metrisable, but only in Flagg's formalism does this result extend to an equivalence of categories between the usual category of topological spaces and the category of all continuity spaces with (classically defined $\epsilon - \delta$) continuous functions.

With these remarks answering precisely which aspects of metric spaces deeply depend on the specifics of $\mathbb R$ is (I believe) not a very simple question to answer. For instance, many properties of classical metric spaces (e.g., compactness being equivalent to sequential compactness) can be seen to be a result of certain properties of the value quantale $[0,\infty]$, so again not really deeply relying on $\mathbb R$ since other value quantales share these properties.

The value quantale $[0,\infty ]$ does have a special role in the theory of value quantales though I can't (yet) refer to any published work (or preprint). In a nutshell, every value quantale embeds into a suitable type of power of $[0,\infty]$, so every value quantale has a dimension over $[0,\infty ]$ (possibly infinite dimension, of course).