What is the definition of a non-linear estimator? I heard that ratio of estimators is non-linear 
Why don't we consider nonlinear estimators for the parameters of linear regression models?
says that LASSO is a non-linear estimator.

I think LASSO has a solution via matrix multiplication. I don't even think that comment from quote is true.
What is definition of non-linear estimator?

Linear estimators of the regression parameters are defined as linear functions of the outcome, typically denoted by $y$. Nonlinear estimators are estimators that cannot be written this way.
The OLS solution is a linear estimator because $\hat\beta_{ols}=(X^TX)^{-1}X^Ty$, so a linear function of $y$ (some matrix times $y$).
For LASSO regression, there is no such closed-form solution where the parameters are written as a matrix times the $y$ vector.
