I'm having trouble understanding an aspect of this question. When I think of a graph, I think of a set of vertices and a set of edges between different vertices. So if, in $K_{3,3}$, I split an edge into two edges by adding a vertex $v_0$ into the 'middle' of the edge and view the new graph as having one additional vertex and one additional edge, that's okay.
But I think you want to add the same vertex again. What does that mean? It seems there are two interpretations: either we literally claim for a moment that our vertex $v_0$ is now also some vertex $v_1$, then we have changed nothing. We have added no edges, we have changed no connectivity, and our new vertex set hasn't really changed.
The other interpretation is to copy the vertex and its connections, and add the copy. Let me give an example to say what I mean. We might start with two points and an edge between them. We add the midpoint, so we have a 2-segment straight line. If we add a 'copy' of the midpoint and its edges, then we get a diamond.
In both cases, if they are called 'subdivisions' (and I would not call the first interpretation a subdivision, but instead the exact same graph), they preserve non-planarity.
All this is to say that no, the vertices don't have to be distinct, but they might as well be.