Thank you, that was fun!

The zeros of the analytic function are where u=0 and v=0. Because u and v are harmonic u=0 and v=0 are one-dimensional curves. So your observation boils down to the observation that u=0 and v=0 only cross at points and never coincide.

It's been, like, 25 years, but aren't the level curves of u and v orthogonal? In that case, they can only ever cross at points.