Arden's lemma applicability on context free grammars
First of all, let me mention that Arden's lemma applies not only to regular expressions but to any equation among languages of the form $r = sr+t$, where $r,s,t$ are arbitrary languages. The smallest solution to this equation (also known as the least fixed point) is always $r = s^*t$. If $\epsilon \notin s$, then this is in fact the only solution. (Otherwise, $\Sigma^*$ is also a solution.)

The language generated by the rule $S\to aSa|a$ is not $a^*$ but rather $a(aa)^*$: only odd powers are allowed. So it seems that the student notes contain a mistake, though in this particular case the calculation yields the correct conclusion since $\epsilon \in L(A)$.

On the other hand, $S\to aSa|a$ and $S\to aaS|a$ do generate the same language.
More generally, if the only rules involving $S$ are $S \to ASA|A$, then replacing these rules by $S \to AAS|A$ or $S \to A|SAA$ result in the same language, $L(A)(L(A)^2)^*$; and if $\epsilon \in L(A)$, we can further simplify this to $L(S) = L(A)^*$.