Can anyone explain why voltage waveform gets copied from output to drain but not current?
Nothing gets "copied", that's not how this works.
What does happen is that the FET feeds a distorted / non sinusoidal current into an impedance. That impedance is formed by L1, C1 and L3 and C3. Since you use ideal components for these (no series or parallel resistance) the impedance behaves as a bandpass filter. This means it will be a high impedance for a narrow frequency band. Other frequencies like harmonics are suppressed.
The high value of the impedance in the narrow pass band is also unrealistic. To make a circuit with realistic behavior you need to add series resistors to the inductors and the capacitors. Parallel resistors are usually not needed.
Why is the impedance important? Because the current coming out of the FET is multiplied by the impedance and that makes the output voltage: Vout = Z(f) * Iin