Is it possible to build push-pull amplifier using simple N-Chanel FETs?

You need two FETs and a BALUN at the input, and a BALUN at the output as in the example attached.
At microwave frequencies can be used microstrip BALUNs.

Vfone, can you explain why they bothered with T2, C19,20,21,22, when connecting the centre tap on T3 to the 48V line, would have worked just as well?
Frank

Compared to single-ended PAs, a push-pull power amplifier has lower input/output isolation, and also about 4 times higher output impedance. All of these conduct to poor stability using conventional BALUNs.
The PA it will work somehow without T2 and DC blocking caps (and feed the drains through T3), but will be hard to make it stable vs frequency and vs various loads.
In 60's Motorola introduced for the first time the concept of adding an extra phased transformer at the output (to feed the active devices) for improving impedance matching and stability.

Vfone, can tell me the source of this picture?

At hight frequency it become quite relative.Class A, B, AB, C and so on. It work till works transformers. It depends from bias it can works like A, or B classes.
Match better are LDMOS transistors at these application.

RSGB Handbook