Find and classify the critical points of the system $x'=2-xy, y'=2x-y$

Question: find the critical points of the system and their stability.

$\frac{dx}{dt}=2-xy$

$\frac{dy}{dt}=2x-y$

Solving this system where each equation is set to $0$, I receive critical points of $(1,2)$ and $(-1,-2)$. To find the stability of each point, I was taught to find the eigenvalues of the matrix where the entries are the coefficients of $x$ and $y$, and that the stability depends on whether the eigenvalues are real, distinct, etc. However I'm not sure what to do in this case, as $\frac{dx}{dt}$ does not have $x$ and $y$ separated.

You have the right idea, but making the matrix from the coefficients of $x$ and $y$ is an oversimplication that, as you noted, fails in this case.

What you're really trying to do is construct the Jacobian by taking the derivative of each equation. Note that for a linear system, the derivative is just the coefficients, hence the simplification. For the system you provided:

$$J=\left[\begin{matrix}\left(\frac{{\rm d}x}{{\rm d}t}\right)_x& \left(\frac{{\rm d}x}{{\rm d}t}\right)_y\\ \left(\frac{{\rm d}y}{{\rm d}t}\right)_x& \left(\frac{{\rm d}y}{{\rm d}t}\right)_y \end{matrix}\right]= \left[\begin{matrix}-y&-x\\2&-1 \end{matrix}\right]$$

From there you can plug in the critical points $(x,y)$ that you found above and find the eigenvalues to determine the type of critical point.