I am wondering if maximum likelihood estimation ever used in statistics.
Certainly! Actually quite a lot -- but not always.
We learn the concept of it but I wonder when it is actually used.
When people have a parametric distributional model, they quite often choose to use maximum likelihood estimation. When the model is correct, there are a number of handy properties of maximum likelihood estimators.
For one example -- the use of generalized linear models is quite widespread and in that case the parameters describing the mean are estimated by maximum likelihood.
It can happen that some parameters are estimated by maximum likelihood and others are not. For example, consider an overdispersed Poisson GLM -- the dispersion parameter won't be estimated by maximum likelihood, because the MLE is not useful in that case.
If we assume the distribution of the data, we find two parameters
Well, sometimes you might have two, but sometimes you have one parameter, sometimes three or four or more.
one for the mean and one for the variance,
Are you thinking of a particular model perhaps? This is not always the case. Consider estimating the parameter of an exponential distribution or a Poisson distribution, or a binomial distribution. In each of those cases, there's one parameter and the variance is a function of the parameter that describes the mean.
Or consider a generalized gamma distribution, which has three parameters. Or a four-parameter beta distribution, which has (perhaps unsurprisingly) four parameters. Note also that (depending on the particular parameterization) the mean or the variance or both might not be represented by a single parameter but by functions of several of them.
For example, the gamma distribution, for which there are three parameterizations that see fairly common use -- the two most common of which have both the mean and the variance being functions of two parameters.
Typically in a regression model or a GLM, or a survival model (among many other model types), the model may depend on multiple predictors, in which case the distribution associated with each observation under the model may have one of its own parameter (or even several parameters) that are related to many predictor variables ("independent variables").