Assuming you mean you want them to be individual objects, and not references to the same object pass the source dictionary into the destination’s constructor:
Dictionary<string, string> d = new Dictionary<string, string>();
Dictionary<string, string> d2 = new Dictionary<string, string>(d);
“so that they are not the same object.”
Ambiguity abound – if you do actually want them to be references to the same object:
Dictionary<string, string> d = new Dictionary<string, string>();
Dictionary<string, string> d2 = d;
(Changing either d or d2 after the above will affect both)