As far as I can see the design you posted isn't for a push-pull converter, it's just a transformer-coupled charge pump, which is very different from typical SMPS which use inductors for energy storage. The circuit has a fixed duty cycle of 50%, and just functions as a voltage multiplier. The reason it can work with that circuit is because the ratio between the two outputs is an integer, so getting +15 and -7.5 should also be possibly with the same approach. But keep in mind it has no feedback or control, so there's no regulation of the output voltages.
It's possible to get two outputs off of a single flyback winding, if the winding has at least one extra tap, but the output voltage will depend on the duty cycle and load, like FVM states, and thus some amount of feedback is likely required.