What is 'spectrum' anyway?
A spectrum is a range of frequencies that is emitted or absorbed by a something.
How can multiple frequencies exist in one signal
The can.
The OP seems to think that all waves are electro magnetic. They are not. Most signals in wires are simply electric.
A frequency is something that repeats over time. However it is not defined by a single voltage point at any instant. The lowest frequency is defined as the period where a signal describes a complete sequence, such that you could take that period, shift it in time by exactly one period and the past wave shape would line up exactly with the future wave shape.
That is just the lowest frequency or fundamental. If that is a sin or cos shaped wave that is all there is to it, if not there is more than one frequency present in that signal. Any wave shape can be created by a sum of sin and cos waves at different amplitudes and frequencies. These frequencies are harmonically related to the fundamental being integer multiples of it.