Suppose $P_1 B_1 P_2 = B_2$. Then $B' = P_1 B_1 P_1^{-1}$ is the adjacency matrix of a graph isomorphic to the first graph, and $P' = P_1 P_2$ is another permutation matrix; we have $B' P' = (P_1 B_1 P_1^{-1}) (P_1 P_2) = B_2$.
The matrix multiplication $B' P'$ permutes only the columns of $B'$. So the question is: when does permuting the columns of an adjacency matrix give the adjacency matrix of a different, non-isomorphic graph?
To understand this deeply unnatural operation, it helps to think of our graphs as directed graphs that merely happen to be symmetric: whenever there is an arc $u \to v$, there is also an arc $v \to u$. The directed adjacency matrix $A$ (which coincides with the undirected adjacency matrix, under this interpretation of all graphs as directed graphs) has $A_{ij} = 1$ whenever there is an arc $i \to j$. Permuting the columns of the matrix is a directed graph operation: we pick a permutation $\sigma$ of the vertices, and permute the ends of every arc in the graph according to $\sigma$, leaving the starts where they are.
It is rare for this to produce another undirected graph (because in general, permuting the columns of a symmetric matrix does not leave it symmetric). But there are examples where it happens.
One nice place to look for these examples is in circulant graphs. An $n$-vertex circulant graph is specified by several values $a_1, a_2, \dots, a_k$ between $1$ and $\frac n2$. The vertices $1, 2, \dots, n$ are arranged in a circle, and two vertices are adjacent if they are $a_i$ steps apart around the circle for some $i$.
Here, if $n$ is even and $\frac n2$ is not one of the $a_i$, let $\sigma$ be the permutation that swaps opposite vertices, and permute the columns of the adjacency matrix according to $\sigma$. We turn a circulant graph with parameters $a_1, a_2, \dots, a_k$ into another undirected graph: a circulant graph with parameters $\frac n2 - a_1, \frac n2 - a_2, \dots, \frac n2 - a_k$. Sometimes this is an isomorphic graph to the original, but usually it's not. For example, the cycle graph $C_6$, a circulant graph with $k=1$ and $a_1 = 1$, turns into a circulant graph with $k=2$ and $a_1 = 2$, which is the disjoint union of two $C_3$'s.
Best Answer
This is not a counterexample. We have $$\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}^{-1} \begin{bmatrix}1 & 1 \\ 1 & 0\end{bmatrix}\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix} = \begin{bmatrix}0 & 1 \\ 1 & 1\end{bmatrix}.$$
The expression $P^{-1}A_2 P = A_1$ is just a formal way of saying "we can get one matrix from the other by permuting the rows however we like, then applying the same permutation to the columns", which is certainly true of any kind of graph isomorphism.