How is the energy conservation principle preserved in the case mentioned below?
Power is homogeneous to a second degree quantity. When the data is doubled, the power is quadrupled.

The energy here (classicaly) is defined in an Hilbert space as a squared $L_2$ norm, deriving from some inner product $\langle\cdot,\cdot\rangle$, assuming a real Hilbert space,  $\|x\|^2 = \langle x,x\rangle$. Hence:  $$\|x+y\|^2 = \langle x+y,x+y\rangle = \|x\|^2 + 2\langle x,y\rangle+  \|y\|^2 \,.$$

Now set $y=x$, you gracefully get $\|x+x\|^2 = 4\|x\|^2$.