  We establish a novel relation between the cluster categories associated with
marked surfaces and the topological Fukaya categories of the surfaces. We
consider a generalization of the triangulated cluster category of the surface
by a $2$-Calabi-Yau extriangulated/exact $\infty$-category, which arises via
Amiot's construction from the relative Ginzburg algebra of the triangulated
surface. This category is shown to be equivalent to the $1$-periodic version of
the topological Fukaya category of the marked surface, as well as to Wu's Higgs
category. We classify the cluster tilting objects in this extriangulated
cluster category and describe a cluster character to the upper cluster algebra
of the marked surface with coefficients in the boundary arcs. We furthermore
give a general construction of $2$-Calabi-Yau Frobenius extriangulated
structures/exact $\infty$-structures on stable $\infty$-categories equipped
with a relative right $2$-Calabi-Yau structure in the sense of Brav-Dyckerhoff,
that may be of independent interest.
