In his MIT 18.06 course, when presenting the "column" view of linear algebra, Strang draws vectors as shown below:
I am assuming that the axis shown on the blackboard (which I have highlighted) are not $x$ and $y$, but an arbitrary system for depicting vectors. Am I correct?
You are correct: the horizontal axis represents the first coordinate and the vertical axis the second, so the vector depicted on the board is $\begin{bmatrix}2\\-1\end{bmatrix}$, the first vector in the equation to the left of the graph in the image. This is akin to the Argand diagram for complex numbers.