How RF signal appears on the gate pin of series-feedback microstrip oscillator?

Is it realted to gate capacitance or maybe somehow to S21 of device? There is no current goes through gate, although we can take poweful RF output on gate pin of series feedback oscillator. So how this thing is called?

Not S21. Maybe you meant S12?

Think also the "Miller Effect". How the equivalent input capacitance of an inverting amplifier gets whatever little feedback capacitance existing between the input and output terminals amplified by (1+Gain).

For an oscillator, there has to be some way a portion of the output gets to wiggle the input gate relative to source, or vice-versa, or maybe a bit of both, not even counting the Miller effect. What you see may not be gate capacitance related.

Also, AC current is involved in driving the gate. For a gate voltage to vary at all, electrons must charge it and discharge it. A gate cannot take zero energy!

We do not know your circuit, but oscillator circuits commonly do have high level signals at all parts, and the only thing that limits the level is the loss of loop gain as the device is driven into non-linearity, or it runs out of supply voltage headroom. The high signal level at the gate could be intentional and inevitable. Also, the attempt to measure it might well have modified it somewhat.