the difference between $\hat{e}^2_i$ and $\sigma^2_i$
Certainly the true variance is not equal to the squared residual.  

In the case or robust standard errors, it has been proven that if we use $\hat{e}^2_i$ we obtain a consistent estimator of the variance-covariance matrix as a whole, even though $\hat{e}^2_i$ is not equal to $\sigma^2_i$ and even though  each single $\hat{e}^2_i$ does not converge in probability to $\sigma^2_i$ (it converges to the true squared error, which is a random variable). 

The fundamental reason for this result is that 

$$E(\hat{e}^2_i) \to_{n \to \infty} E(e^2_i) =\sigma^2_i$$

So the OP's phrase 


  When we calculate the robust robust standard error, we simply repalce
  $\sigma^2_i$ with $\hat{e}^2_i$. Thus, theses two are considered to be equal.


is simply wrong.