The category of vector (over $k$) representations of a group $G$ is isomorphic to the category of left $k[G]$-modules, where $k[G]$ is the group ring. This isn't just an equivalence of categories, but rather an isomorphism, which is stronger. However we define a tensor product of representations as $g(v\otimes w)= gv\otimes gw$, which differs from the tensor product over $k[G]$ of $k[G]$-modules. For various reasons, we have to define it this way. For starters, one cannot tensor two left modules if the base ring is not commutative.
But there's something unsettling about using a different tensor, about forgetting the base ring. Is there some deeper explanation for this discrepancy other than just "it works"?