For the Cantor starcase function $f:C\to[0,1]$ from the standard ternary Cantor set $C$ onto the interval $[0,1]$ and for the standard Euclidean metric $d$ on $C$ the quotient pseudometric $d_\sim$ is constant zero (this follows from the fact that the Cantor set $C$ has length zero). So, the pseudometric $d_\sim$ is not necessarily a metric.

This example should be known but I cannot mention a suitable reference at the moment.

For the Cantor starcase function $f:C\to[0,1]$ from the standard ternary Cantor set $C$ onto the interval $[0,1]$ and for the standard Euclidean metric $d$ on $C$ the quotient pseudometric $d_\sim$ is constant zero (this follows from the fact that the Cantor set $C$ has length zero). So, the pseudometric $d_\sim$ is not necessarily a metric.

This example should be known but I cannot mention a suitable reference at the moment.

Added in Edit. Essentially the same counterexample is discussed in the answer of Wlodzimierz Holsztynski to this MO-question.