Complexity of Multi-colored Clique when every color pair induce biclique+isolated vertices
The problem is polynomial-time solvable.
Observe that we have to pick exactly one vertex from each color class to form a $k$-clique.
If there is a vertex $x \in V_{i}$ that has no neighbor in some $V_{j}$ with $i \ne j$,
then no $k$-clique includes $x$. Thus, we can remove such a vertex.
Repeat this until every vertex has a neighbor in each color class (except for the class it belongs to).
If some class becomes empty, then the graph does not have a $k$-clique.
Otherwise, since the property ``every pair of color classes induces a biclique + isolated vertices''
is closed under taking induced subgraphs,
the remaining graph is a complete $k$-partite graph, which contains a $k$-clique.