Common mode and differential mode
We analyse signals as common mode and differential when it makes sense to do so, when it simplifies the maths, or simplifies the intuition about the circuit. You are correct that for general signals, the concept is not very useful.

So, when is it useful to use differential and common mode concepts?

When analysing a differential input amplifier, like an op-amp, or a long-tailed pair. Such amplifiers usually have radically different gains to differential signals and to common mode signals.

When analysing noise on a signal that has been transmitted differentially. Balanced differential signals are very common in electronics. They are easy to generate with a transformer, or differential output device. They are used as, in general, interference during transmission gets picked up on both conductors, more or less equally, so results in only a common mode signal, with no differential. If they are now received by an amplifier with a good 'common mode rejection ratio' (the normal way the difference in gains is expressed), then the noise will be reduced with respect to the signal.