As we have seen, a musical tone is made up of a set of related harmonics, which give the sound it's character. 'Formants' are a more human-oriented way of examining sound, and of understanding how we hear it.
While no to voices and no two pianos are identical and therefore when examined technically on a computer screen, the actual shape of the waves produced by two various singers or two different pianos will probably look totally different. But if we hear these sounds, even very briefly, we will be easily both recognise a voice as a voice, and furthermore recognise one individual singer or speaker from another. If the sample we hear is speech, we will be able to additionally identify the various sounds of speech (phonemes), and of course the words themselves.
This is because of Formants. Formants are patterns of harmonics which the ear can easily recognise. The ready distinction we make between spoken vowels for example is because of each vowel having different Formants. So although two vocal timbres may be different (therefore technically different waveforms), the two voices will both contain the characteristic pattern of harmonics that identifies the sound as human voice saying a particular phoneme. Much like we can identify a rectangle or a triangle in a generic sense because of its shape, regardless of size or proportion, colour or other attributes. .
For the technical: Formants are frequency peaks that are constant regardless of the fundamental pitch.
The formants present in a speaking voice are strongly variable and quite discontinuous. They are very strongly dependent of the vowel or phoneme. The formants of the singing voice are somewhat different. There is a strong underlying formant called, appropriately, the 'Singers' Formant.' This group of frequencies underlies the formant pattern of the voice, and is varied somewhat by the vowel being voiced. Overall the formants of the singing voice are less variable than those of speaking, with smoother transitions and slightly lower overall average frequency.
Some speakers' voices do contain a steady 'Speakers' Formant' in addition to the vowel-based formants, but this is not a permanent attribute of the speaking voice and plays less part in the overall spoken sound than the Singers' Formant plays in the sound of the singing voice.