Is it possible for the symmetric squares of a pair of non-isomorphic curves $C_1, C_2$ defined over a field $K$ to be isomorphic?

EDIT: as the user @abx has mentioned, there exist such examples in the case of genus 2 curves. I would like to add further restrictions which correspond to the special case I am interested in: what if $C_1$ and $C_2$ are both smooth plane quartic curves?