What is this output stage?
Neil has already discussed some things. I'll add one additional comment which is more directly related to the circuit you show and it's more common function within a larger system. (And yes, it's push-pull arrangement just as Neil and Rohat have already mentioned.)

In the 1950's HAM radio operators using their cars as a base station and both car and farm radio receivers had to operate off of "DC car batteries" for power. They often used a mechanical "vibrator" (see here for a simpler example of one of these devices), together with a transformer, in order to generate the higher plate voltages required by vacuum tube circuits at the time. In the late 1950's and after, lower voltage vacuum tubes and transistors took over for much of the radio receiver part of the market. HAM radio, of course, requires quite substantial plate voltages and that continued the need for these mechanical vibrators for a while.

When I was growing up, I'd find myself replacing worn out mechanical vibrators with new ones not frequently, but also not infrequently. I also scavenged them from wrecked cars of the day.

A very nice video from Mr. Carlson's Lab provides an excellent way to see how this all works. I've set the video to start with the mechanical vibrator, itself. You can go earlier in the video where the schematic is discussed to see this kind of circuit in no less than two places in this schematic. One of them is for the power supply itself and uses the mechanical vibrator I mentioned. Another is in the upper-right corner of the schematic where Mr. Carlson will soon place his fingers when talking about the class-AB amplifier stage, as well. This video is an excellent source of information related to this kind of stage you have, which uses similar ideas to perform two completely different functions.

So, from the above video and schematic, I think you can see that this concept is not only used in class-AB stages where a transformer may be required to match impedances better, but it is also used in power supplies, too. And that's just two uses.

The basic idea is to turn on one side, then turn that off and turn on the other side, in alternating fashion. With the positive voltage rail at the center-tap of the transformer, this treats the center-tapped primary as two separate and similarly sized primary coils, operated in alternately opposing directions, just as an AC source would do with a single primary coil. There will be different design considerations if it is being used to drive a power supply transformer vs a class-AB push-pull transformer impedance-matching driver circuit. But the basic ideas are close enough to be grouped.

It's push-pull, just as you say.