Signal variance and power connection
The terminology is a bit unfortunate here. &quot;Signal Power&quot; and &quot;Physical Power&quot; are two very different things.

why is E(x(n)2) called signal power?

Because signal power is defined this way.

Is it really power? Any proof?

It's NOT physical power. So there is nothing to proof here. A signal doesn't have power in the physical sense. Physical power only occurs when the signal is associated with some sort of physicals arrangement with an impedance in it. Physical power typical is the product of two complimentary physical field quantities (Voltage/Current, Force/Velocity, etc).
We can relate Signal Power to Physical Power:

It never DIRECTLY represents physical power. The units are the square of the original units and for it to be physical power the units of x(n) would have to be $\sqrt W$ which doesn't exist
It's often PROPORTIONAL to power (or intensity). That's the case if the impedance across or through which x(n) occurs is real and frequency independent.

Examples:

Voltage across an ideal resistor: $&lt;x^2&gt;$ is proportional to the power dissipated in the resistor
Sound pressure measured with a pressure microphone sufficiently far away from a sound source: $&lt;x^2&gt;$ is proportional to the intensity of the sound field at the location of the microphone
Voltage across a dynamic loudspeaker: $&lt;x^2&gt;$  does not represent anything that can easily be  related to physical power.