Can you differentiate between the sounds of sinusoidal, saw-tooth and square waves? Only the sinusoidal wave is monotonic. It truly consists of one single pitch. The saw-tooth and square waves are actually the superposition of many sinusoidal waves of different frequencies. So they contains higher harmonics on top of the fundamental. While they all sound like the same pitch to us, they also sound different because of the different (amplitudes) of the harmonics they contain.
The sound produced by the guitar (string) and the recorder (open pipe) are also not monotonic. It takes a bit of imagination to realize that the standing waves formed on the string and the air in the pipe actually consists of many frequencies at the same time. But that’s what allows the sound produced by different instruments to have different timbres.