Why do ratios of these Fibonacci-type sequences approach $\pi$?

The correct ratio is $\frac{6\phi^2}5=3.14164079$, which is remarkably close to $\pi$. The $6/5$ comes because the starting values are $6,6$ in one and $5,5$ in the other. The $\phi^2$ comes because the sequences are shifted two steps from each other - both 6's appear before the 5's.

Recurrences of the form

$$T_n=T_{n-1}+T_{n-2}$$ are linear and known to have a general solution of the form $$T_n=C_0z_0^n+C_1z_1^n,$$ where $z_0,z_1$ are the roots of the "characteristic equation", $z^2=z+1$.

By the usual formulas,

$$z_0,z_1=\frac{1\pm\sqrt5}2.$$ Using the initial conditions,

$$T_0=C_0+C_1,\\T_1=C_0z_0+C_1z_1.$$

As $|z_0|>|z_1|$, the first term quickly dominates and

$$T_n\approx\frac{T_1-z_1T_0}{z_0-z_1}z_0^n.$$

In your case, for large $n$,

$$\frac{A_n}{B_n}\approx\frac{A_1-z_1A_0}{B_1-z_1B_0}=\frac{3\sqrt5+9}5=3.1416407864999\cdots$$