Prove np-hardness of dividing items from the lists
To prove that a problem is NP-hard, we need to reduce an NP-hard problem to it (which in this case as you mentioned would be 3-coloring).

So what we are trying to do, is to map every graph $G(V,E)$ to an instance of your problem, where every node $v_i$ will be mapped with an element $el(v_i)$ and every edge $\epsilon\{v_1, v_2\}$ would be represented with a two-element list $\{el(v_1), el(v_2)\}$.

So now a coloring $f:V\rightarrow C$ would be reduced to a coloring on the set of elements in your problem which means, if we found a coloring on the elements of the proposed problem, we can use the same coloring on our graph and it would be a proper coloring.

Proof.
If $f:V\rightarrow C$ isn't a proper coloring, that means there is an edge with both nodes having the same color, which means there is a set with two elements and both of them have the same color which contradicts the statement of your problem.