Q1. Is the union of matrices $A$ and $B$ denoted by $A|B$?
The following example is taken from a monograph by M. Hazewinkel entitled "Handbook of Algebra", Volume 6 Published by North Holland, (2009) ISBN: 0444532579,9780444532572.
Example 6.29. Let us consider the following two matrices $ A=\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0& 0 & 1 \\ 0 & 1 & 0 \end{bmatrix} $, $ B=\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{bmatrix}.$ Then it is straight forward to see that $r(A)=2$, $r(B)=1$. Then it holds that $r(A|B)=4>3=r(A)+r(B)$.
Q2. What is the union of two matrices?
According to what i think about a matrix union, the union of the given mattrices should have been $$A|B=\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1& 0 & 1 \\ 1 & 1 & 0 \end{bmatrix} $$ Therefore, its rank should have been $r(A|B)=3$ but not $4$