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$\begingroup$OH you are right; Indeed it is! Peculiar ! For me it was the set of mobius transformations that commute with $c(z)=\frac{-1}{\bar{z}}$ and writing $c$ as $c[z,w]=[-\bar{w},\bar{z}]$, we can see where does $U(2)$ come from. $\endgroup$
Identifying your points with matrices $M$ with column vectors $(a,b)^T$ and $(c,d)^T$, your equations come from the components of $M M^\dagger=I$ where $M^\dagger$ denotes the conjugate transpose. So $C$ is $U(2)$ and $S$ is $SU(2)$. Since $SU(2)$ is diffeomorphic to $S^3$, it is orientable.
$\begingroup$Yes, but it hadn't been answered yet. Also, it is unclear to me which appeared first, although I gather you received notifications regarding the comments. Now at least the question appears as answered.$\endgroup$