In theory, something magical does indeed happen with the last .00001 degree step, namely you lose the ability to derive arbitrary phase angles from that source.
In practice it becomes harder and harder to derive other phase angles the closer you get to 180, and at some point you would not make the effort.
Definitions again. If there are more than two phases present in a circuit then you have multiple phases. this can apply to both voltages and currents.
Inside the two wire transformer circuit, there is only one phase present. The same electrons, fields, etc. even at the atomic level. No matter which way you take the two terminals.
Inside the three wire transformer circuit, there can be multiple phases present. A different set of electrons, fields, etc. Now it matters how you define the system of currents and voltages.
Define them one way and they are twin sets so would not be counted as unique. This uses two different reference points for the two waves but when shifted and overlaid they "match".
Define them a different way and they are twin sets so could be counted as unique. This uses one reference point (positive away from center) for the two waves and they are opposites.
The two transformer halves do not have to act the same. We usually use the full winding or one of the halves for a circuit. A circuit that uses both halves (rectifier, pulse, etc.) can have different electrons, fields, etc. Indeed, one half may have load current and the other have none. In the rectifier/pulse case, one half sends a positive signal but the other half conducts nothing, then 180 degrees later the other half sends a positive signal while the first half conducts nothing.
add: cut & paste typo.