Two simple graphs are isomorphic if there is some bijection between the vertex sets and edges are preserved under this mapping. Can this also work for multigraphs? I think that the same definition can be used because we are only adding more edges.
Yes and no.
I assume that by "multigraph" you mean a graph where you allow more than one edge between each pair of vertices.
In that case you can certainly say that $G$ and $H$ are isomorphic iff there is a bijection $f:V(G)\to V(H)$ such that for every $v,u\in V(G)$, the number of $H$-edges from $f(u)$ to $f(v)$ is the same as the number of $G$-edges from $u$ to $v$.
However, when you speak about an isomorphism between $G$ and $H$, you would usually want that to be a combination of a function that maps vertices to vertices and a function that tells exactly which edges in $G$ maps to which edges in $H$. For example if $G$ and $H$ are both directed graphs with $2$ vertices and exactly $2$ parallel edges from one vertex to the other, then there would be two different isomorphisms between $G$ and $H$, depending on which edges map to which.