Is there a conceptual reason why topological spaces have quotient structures while metric spaces don't?

You can define a (pseudo)metric on a quotient of a metric space. Let $X$ be a metric space with metric $d$ and an equivalence relation $\sim$. Say that a chain between two points $x,y\in X$ is a sequence of points $x=a_0\sim b_0$, $a_1\sim b_1$, $\ldots$ $a_n\sim b_n=y$, and define the length of such a chain to be $\sum d(b_i,a_{i+1})$. We can now define the distance $d([x],[y])$ between two equivalence classes to be the infimum of all lengths of chains from $x$ to $y$.

It's easy to see that this is a pseudometric on $X/{\sim}$ (a metric where the distance between two distinct points might be $0$). This descends to a true metric on the quotient $Y=X/{\sim}'$, where $x\sim' y$ if $d([x],[y])=0$. Furthermore, $Y$ can be characterized by the following universal property: (non-strictly) distance-decreasing maps from $Y$ to a metric space $Z$ are naturally in bijection with distance-decreasing maps from $f:X\to Z$ such that $f(x)=f(y)$ whenever $x\sim y$.

More generally, a similar construction shows that the category of metric spaces and distance-decreasing maps has all connected colimits (colimits over connected diagrams). If you generalize metrics to allow the distance between two points to be infinite, you can construct all colimits, and also all limits (use the sup metric on products).

One way to get around the lack of quotients is to work in the category of Uniform Spaces. This is more general than a metric space, but still gives you a way to talk about how "close" two points are. Uniform spaces are also cool because they simultaneously generalize metric spaces and topological groups. I believe their study was popular back in the 60s and 70s for instance because this category of uniform spaces has much nicer categorical properties than either of the categories it generalizes. For instance, with uniform maps it's complete and cocomplete, as can be seen here.

Even if your maps are just continuous maps, you still get quotients as can be seen here. Isbell's book is a great reference for this subject.

Side note: I like Eric Wofsey's answer a lot and I wonder if it's secretly related to this answer. The wikipedia page points out the connection between uniform spaces and pseudo-metrics but for categorical things it's probably easy to work with the former than the latter.