Error analysis for uniform quantization of uniform input
This assumes that both quantization error and original signal are uniformly distributed and have zero mean.

That's a good assumption for quantization noise, but it feels very arbitrary for the signal.

The variance will be proportional to the power (not energy), ONLY if the signal is mean free. Power is defined roughly as the "mean of the squares"  and variance as the "mean of the squares after subtracting the signal mean".