It looks like he's using the term "regular representation" to mean: a representation of $A \times G$ on $L^2(G; H)$ constructed by combining a given representation of $A$ on $H$ with the left action of $G$ on $L^2(G;H)$. If you start with $A = {\bf C}$ then there is "only one" representation of $A$ to start with, so there is only one regular representation of ${\bf C}\times G$. (Not technically true, and maybe this is your point, since we can actually represent ${\bf C}$ on any Hilbert space $H$, but all that changes about the construction is to tensor everything with $H$.)

It looks like he's using the term "regular representation" to mean: a representation of $A \rtimes G$ on $L^2(G; H)$ constructed by combining a given representation of $A$ on $H$ with the left action of $G$ on $L^2(G;H)$. If you start with $A = {\bf C}$ then there is "only one" representation of $A$ to start with, so there is only one regular representation of ${\bf C}\rtimes G$. (Not technically true, and maybe this is your point, since we can actually represent ${\bf C}$ on any Hilbert space $H$, but all that changes about the construction is to tensor everything with $H$.)