I think it's not that simple. A tuner is "hearing" the fundamental and all of the harmonic overtones combined. It has to guess at which frequency is the fundamental, even if the overtones are actually stronger than it amplitude-wise, and it does that by looking at the nature of the repeating overtone pattern and extrapolating back to the fundamental.

I think you can end up an octave too low (half the actual frequency) if the waveform repeats in a way that implies a different overtone repetition pattern, for example if there's an every-other-cycle artifact to the waveform.