In a split phase system there are 6 possible voltage measurements which can be made. As mivey has said, all of these voltages are 'real'. None are more real than the others. They can be measured at the panel, they can be measured at the transformer. One doesn't have to measure them at all, one doesn't have to risk getting splintered climbing the pole.
One is free to represent these voltages any way he wishes. I choose to represent them as phasors which are just complex numbers. I also choose the neutral as a reference because in so doing the phase angle indicates the phase displacement of the two waveforms.
Now I write the expressions for the fixed phasors, 120Vrms @ 0 and 120Vrms @ PI. I could write -120Vrms @ 0, but that is awkward and does not show the phase angle.
I have done all this without any concern for the real world connections. I know what they must be, therefore I believe they are. Nuf sed!
Furthermore, to answer the OP's question, we can certainly assume ideal situations. No need to bother with impure sinusoids, noise, loading, harmonics, energy flow, phase shift vs inversion, non-linear loads, unbalanced loads, skin effect, teeter-totters, etc., etc., etc.
Now we can say, "Although there are technically two phases present, we call it a single phase system because it is supplied from one phase of a three phase system--usually'. That's all folks.