The first thing to notice is where $2y$ comes from. You have two values at $x=0$. But for symmetry reasons, they differ only in sign. Hence, they're $+y$ and $-y$, and their difference is $2y$. Similarly, your Y signal has an amplitude A, and thus can have range from $-A$ to $+A$. The difference is $2A$, which you call $B$. Therefore, $2y/B$ is the ratio of the Y signal at $x=0$ and the max Y signal.
But why is this important? Let's look at some special cases. If the phase shift is zero, your ellipsis becomes degenerate. It's a line, and $y=0$. If the phase shift is $\pi/2$, it's a circle and $y=B/2$. Clearly the shape matters, and $y/B$ is a measure of that shape.
Not let's put in numbers:
$$x = A \sin(ft)$$
$$y = A \sin(ft+\phi)$$
$x=0$ means $\sin(ft) = 0$ and therefore $y = A \sin(\phi)$. Substituting $B$, you get $$x=0 \implies y = B/2 \sin(\phi) \implies \sin(\phi) = B/2y\qquad \qquad\Box$$