  Given an edge-coloring of a simple graph, assign to every vertex $v$ a set
$S_v$ comprised of the colors used on the edges incident to $v$. The
$k$-intersection chromatic index of a graph is the minimum $t$ such that the
edge set can be properly $t$-colored, additionally requiring that for every two
adjacent vertices $u$ and $v$, $|S_u \cap S_v| \le k$. For all $k \neq 2$, this
value is known for subcubic planar graphs, and furthermore, these values are
best possible. We naturally extend this definition to multigraphs with bounded
edge multiplicity, and we show that every subcubic planar multigraph with edge
multiplicity at most two has 2-intersection chromatic index at most 5, which is
sharp.
