I want to write the statement $A \approx B = C$ in a multi-line format. Is it appropriate to write it as
\begin{equation} \begin{split} A &\approx B \\ &= C \end{split} \end{equation} or does this imply that $A = C$? In other words, do multi-line equations read like a book (left-to-right, top-to-bottom) or like a table?
I don't see why not.
We often write
$$\begin{align*} A&=B \\ &\leqslant C \\ &=D\end{align*}$$
and it never means that $A=D$. Only $A\leqslant D$.
But the problem in your case is that if you write
$$\begin{align*} A&\approx B \\ &\approx C \\ &\approx D\end{align*}$$
then you won't necessarily have $A\approx D$ because the relation $\approx$ is not transitive.
We usually reserve this kind of presentation for relations transitive to each other.
That is why we never write
$$\begin{align*} A&<B \\ &=C \\ &>D.\end{align*}$$
which would not make any sense.