Let $x=A|B,x′=A′|B′$ be cuts in $\mathbb{Q}$. Why do we not define $x·x′ = (A·A′)| \text{rest of}\,\mathbb{Q}$?
My intuition was to define $A=A'=\{r\in\mathbb{Q}|r<0\vee r^2<2\}$, as both $x$ and $x'$ are irrational cuts while $x\cdot x'=2^{*}$ is a rational cut. However, even if $2^{*}=C|D$, $C$ still doesn't have a highest element like $A$ and $A'$. So I wouldn't be able to show that $A\cdot A'\neq C$ as I initially intended to disprove the given definition.
Was my intuition in the right direction or am I missing another perspective? I would appreciate any hints/nudges.
The simple reason is that $A\cdot A'$ contains products of pairs of negative numbers. Which is to say, for instance, in your concrete example we have $4=(-2)(-2)\in A\cdot A'$. And that is not desirable.
More generally, $A\cdot A'$ is never bounded above (although it could be bounded below). This is not a good thing for the lower part of a cut.