I'm confused about the precise definition of an inner automorphism of an algebraic group. Here is what Milne says in his book on algebraic groups:

Let $k$ be a field, let $\overline{k}$ be an algebraic closure, and let $G$ be an algebraic group over $k$. Let $Z$ (or $Z(G)$) denote the center of $G$. Then an automorphism of $G$ is said to be inner if it becomes of the form inn$(g)$ over $\overline{k}$, where $g \in G(\overline{k})$. The action of $G$ on itself by inner automorphisms, $(x,y) \mapsto xyx^{-1}$, is invariant under $Z(G) \times e$ acting by translation, and so it factors through $G/Z(G)$. The inner automorphisms of $G$ are exactly the automorphisms of $G$ defined by elements of $G/Z(k)$.

I don't understand why we don't simply say that the inner automorphisms of $G$ can be identified with $G/Z$. I think Milne is identifying $G/Z(k)$ with a subset of the underlying topological space $G/Z$.

algebraicgroup $\mathrm{Aut}(G)$ ofalgebraicautomorphisms—that is, in particular, as an algebraic group—whereas Milne is regarding the inner automorphism group as an abstract group (as @anon points out below). $\endgroup$ – LSpice Sep 9 '18 at 23:56