$E(\mathbb{F}_{k^q})[r]$ is the set of $r$-torsion points, which means all points, $P$ where $rP = O$ (I think).
Correct.
I guess this is isomorphic since there are 4 elements in each set. But... I'm not sure how stating there is an isomorphism adds any value?
For example: We could instead just say $E(\mathbb{F}_{k^q})[r]$ has $r^2$ elements (which is the size of $Z_r \times Z_r$).
Understanding this structure is quite important for a lot of applications in cryptography. For example, it is very fundamental in isogeny-based cryptography. The reason for this, is because as a product of two cyclic groups, it is generated by two (independent) points $P, Q$ of order $r$. That is, every point in the torsion can be written as $[a]P + [b]Q$ for some coefficients $a,b$. Compare this, say, to classical elliptic curve cryptography, where we work in a cyclic group and every point can be written as $[x]G$ for a single generator $G$. There are no points of order $r^2$ in $E(\mathbb{F}_{k^q})[r]$, even if the group itself has order $r^2$.
Because of this structure, there are $r+1$ subgroups of order $r$ in the torsion subgroup. This is important in isogeny-based cryptography because each of those subgroups form the kernel of a different isogeny from the curve $E$.
Studying the structure of the $p$-torsion subgroups when $p$ is the characteristic of the field (what it seems like you have called $k$ - I suspect you wrote $q$ and $k$ the wrong way around) also classifies elliptic curves into "ordinary" and "supersingular" curves.
For more information, see Silverman's "The Arithmetic of Elliptic Curves", section III, Corollary 6.4.
In pairing-based crytography, this structure is also extremely important. A good reference for more info in this area is Craig Costello's "Pairings for beginners". (See chapter 4 especially).
