Group algebra functor preserves colimits

Question

Consider a commutative unital ring $R$ and the group algebra functor $$R[-]:\bf{Grp}\rightarrow {Alg}_R$$ which has the group of units functor $$(-)^\ast:\bf{Alg}_R\rightarrow \bf{Grp}$$ as right-adjoint. Hence it preserves colimits, in particular we should have $$R[G\ast H]\cong R[G]\otimes_R R[H]$$ where $G\ast H$ denotes the free product (which is the coproduct) of groups. However apparently the equality $$R[G\times H]\cong R[G]\otimes_R R[H]$$ holds. So if I am not mistaken we have $$R[G\ast H]\cong R[G\times H]$$ for any two groups $G$ and $H$. Am I correct? It does not seem right to me. Also, is there a way to derive the identity $R[G\times H]\cong R[G]\otimes_R R[H]$ from abstract nonsense?