Being able to sense timbre has nothing to do with being able to tell the shape of the waveform of a sound.

You're able to recognize red, you're unable to tell me it's wavelength (intuitively)

Recognising red _is_ recognising its wavelength. True, we don't know what frequency we hear/wavelength we see is associate with which specific number, but that's irrelevant.

Humans are born with the capability of doing a mechanical Fourier decomposition in our ears. Knowing the amplitude at each frequency, as all humans that can hear do, gives us information about the waveform.

Sensing timbre therefore is information about the waveform. We do not have complete information (e.g. we do not sense all the phase information) but we have approximate information about the time-domain waveform.

To be more specific, lots of high frequency content means we are hearing sharp transients, so quickly changing waves. Lots of low frequency content means we are hearing smoother waves