Currently in Blender there is no option (atleast by default) to simply duplicate shape key.

This addon is adding a button to duplicate currently active shape key.

Addons works on Blender 2.82+ (atleast those are the versions I've tested it on).

Let me know if you find any bugs working with this tool.

To do so you usually go through the same workflow and I idea is to avoid spending time on it:

1) go through all the shape keys and set their value to 0;

2) set value to 1 for the shape key you want to copy;

3) Use "New Shape Key From Mix".

4) Restore all values you had before step 1.

Old workflow: