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).