Correlation length anisotropy in the 2D Ising model
The correlation length of the 2d Ising model has been computed explicitly. You can find the expression in the famous book by McCoy and Wu. Here's a plot of the inverse correlation length (i.e., $1/\xi$) at various temperatures, taken from this recent review paper:
This is only to show the directional dependence, as the radial scale is not the same for all pictures. The temperature decreases from left to right (you can see the isotropy appearing close to the critical temperature) from close to $\infty$ to close to the critical temperature. Below the critical temperature, the behavior is exactly the same, since self-duality of the model implies that, for any $T<T_c$, $\xi(T) = \xi(T^*)/2$ where the dual temperature $T^*=T^*(T)$ satisfies $T^*>T_c$.
You could study this problem near the fixed point (the right two images in Yvan's answer) by looking for the most relevant operator with the right symmetry charge.
For instance for a rectangular lattice we would be looking for spin 2 operators, a triangular lattice spin 3, and a square lattice spin 4.
Because these higher spin deformations in the Ising model are coming from descendant operators, one expects about an order of $(T-T_c)$ separation in the anisotropies between each case.
I don't know how to explain other interesting features though, like why the correlation length has a cusp at low temperatures. That's pretty cool!