No, electr_peter is not suggesting using the existing center-tap wire. As you say, that is only the center-tap for ONE of the secondary windings.
electr_peter is saying to connect the two secondary windings IN SERIES. The point where the blue wire and the yellow wire connect would be the center-tap of the COMBINED secondary windings.
Yes, you could also use separate bridge rectifiers on each of the secondaries, and then wire the outputs into a dual rail, +15 and -15V output arrangement. There are some "audiophool" circuits that use that topology. They think that having MORE components somehow makes the circuit "better". Of course there is no merit in that fantasy, but audiophools are very stubborn and dislike facts.