To replace $z$ by $t$ is arbitrary, the point is that many authors treat linear ODE with the picture in mind of a system evolving in time, therefore the variable is mostly denoted by "$t$" for "time".
Another point is that it is a convention to reduce linear higher order differential equations $ a_n (t) w^{(n)} + ... + a_1(t) w =0 $ to a linear system of differential equations of order one by introducing the "auxiliary" functions $ w_0 (t) := w(t) $ $ w_1 (t) := w^{(1)}(t) $ etc. so that every linear differential equation can be written in the form $ \frac{d}{d t} \vec{w} = A(t) \vec{w} $ with an appropriate matrix $A(t)$ and the vector $\vec{w}(t) := (w_0(t), ..., w_n(t))$. Such a reformulation is always possible, it is equivalent to the original one. You treat the lower order of differentials (1 instead of $n$) for the number of equations ($n$ instead of 1). This formulation is the "standard formulation" for linear ordinary differential equations: It is useful because one can deduce information about the system from information about the matrix $A$, i.e. by using linear algebra, for this reason many authors use mainly this reformulation.
For example, if $A$ is diagonalizable and independent of $t$ (then one talks about an "autonomous" system of ODE), by determining a basis of eigenvectors and formulating the equation with respect to this basis, you immediately get a solution of the equation.