Can a graph have multiple identical elements? Can a graph have multiple nodes that have the same value ? For instance, could a graph holding numbers have the same number present multiple times across the itself ?  

My current approach is to not allow the insertion of the same element multiple times, and to instead enable an element to have an associated value, is it correct ?  

Thank you.
Graphs don't have values.  Graphs don't hold numbers.  You don't insert a value into a graph.  Graphs have vertices and edges.

You can augment a graph to associate each vertex (node) with a number, if you wish.  If you do that, it's up to you to define whether or not you allow two different vertices to have the same number or not.  There is no universal answer.