A single vertex can have multiple colors at the same time, because it can belong to multiple faces (with different colors). The vertex colors are also stored in face loops and not in vertices themselves.

There is no way to set color to single face-loop element in UI, you surely can set vertex color to vertex (which will propagate to all the associated face-loops vertices) by drawing over it in Vertex Paint mode. This results in the color being interpolated over all the surrounding faces:

![enter image description here][1]

But you can limit the drawing only to selected faces, not individual vertices.

You can set the colors to individual face-loop vertices with python like this:

    for ipoly in range(len(ob.data.polygons)):
        for ivertex in ob.data.polygons[ipoly].loop_indices:
            ob.data.vertex_colors["col"].data[ivertex].color = (R,G,B)

With this you can also achieve non-smooth vertex colors (here some vector displacement visualized on faces):

![enter image description here][2]


  [1]: https://i.sstatic.net/HJNzO.gif
  [2]: https://i.sstatic.net/sx7RV.png