TiKZ node name prefixes in scopes

TiKZ node name prefixes in scopes

I am using TiKZ to draw two copies of the same graph above each other, and
now I want to add edges between a vertex in one copy and a vertex in the
other copy. It suddenly occurred to me that it would be nice if I could
put each copy in a separate scope, give each scope a name, and refer to
nodes inside the scope in an "object oriented" way by referring to the
node as <scope name>.<node name>. To illustrate, it would be cool if I
could do something like this:
\documentclass{article}
\usepackage{tikz}
\begin{document}
\tikzstyle{vertex}=[circle,draw,fill=black!20]
\begin{tikzpicture}
% ---- Copy 1
\begin{scope}[yshift=-32pt,name=G1]
\node[vertex] (u) at (0, 0) {u};
\node[vertex] (v) at (0, 0) {v};
\end{scope}
% ---- Copy 2
\begin{scope}[yshift=32pt,name=G2]
\node[vertex] (u) at (0, 0) {u};
\node[vertex] (v) at (0, 0) {v};
\end{scope}
\draw (G1.u) -- (G2.v);
\end{tikzpicture}
\end{document}
Here, I refer to node u of scope G1 as G1.u, and to node v of scope G2 as
G2.v.
Is something like this possible in TiKZ?