For concreteness, let's say that $(X,d)$ is a metric space homeomorphic to $\mathbb{R}^2$ whose Hausdorff 2-measure $\mathcal{H}_d^2$ is locally finite.

We can pass from $(X,d)$ to the length metric, denoted by $\overline{d}$, defined by the infimum of the length of rectifiable curves joining two points in $X$. The inequality $d \leq \bar{d}$ holds in general, so $\mathcal{H}_d^2(E) \leq \mathcal{H}_\bar{d}^2(E)$ for every Borel set $E$. My question is whether this inequality can be strict.

This question does need at least one refinement, however, since a negative answer can arise in an uninteresting way. Namely, you might have an uncountable collection of points in $X$ that are not accessible by a rectifiable curve. For such a point $x \in X$, we have $\bar{d}(x,y) = \infty$ for all $x \neq y$. Taking the definition of Hausdorff measure literally, in this situation the Hausdorff 2-measure of an uncountable set of inaccessible points is infinite. Thus the set of inaccessible points should be ignored. Let $F \subset X$ be the set of inaccessible points. The question is now: does $\mathcal{H}_d^2(E\setminus F) = \mathcal{H}_{\bar{d}}^2(E\setminus F)$ hold for every Borel set $E \subset X$?