Tone and timbre are absolutely key here, but also inflections, vibrato, stresses, breathing, power, in fact there are a huge range of factors that make you sound like you.
The spectral pitch clip you show above demonstrates elements of this - where the narrator mentions fundamental and overtones, the balance of amplitudes of these give the tone or timbre. You can also see the lead in to each note - does the singer hit it exactly or bend up or down to it.
In fact videos like that one are good at answering your question - much better than the answers we can give here. If you wanted to be able to impersonate another singer you would need to be able to produce the same harmonics at the same amplitudes as them by altering your body's resonances (your chest, throat, nose all have a part to play in this) - there are entire training courses on this.