What is a name of regression? Let's say there is a set of independent variables $x_1, x_2, ..., x_n$ and a target variable $y$. A transformation is applied to the initial set of variables: $z_1=f_1(x_1,x_2,..., x_k)$, $z_2=f_2(x_{k+1},..., x_n)$, and then a multiple regression $y=g(z_1, z_2)$ is built.
What is this regression called?

This is called feature extraction.
For example, let $x_1$, $x_2$, $x_3$, and $x_4$ be candidate predictors. Let $z_1=f_1(x_1,x_2)= x_1+x_2$, and let $z_2=f_2(x_3, x_4)=x_3+x_4$.
Then $\hat y=\hat\beta_0+\hat\beta_1z_1+\hat\beta_2z_2$ is a way to for the $x_1$ and $x_3$ coefficients to be the same as the $x_2$ and $x_4$ coefficients, respectively.
This need not be a desirable property, but you might want it in some problems.
It doesn’t take much past this example to get into nonlinear basis functions for linear regression. I like this video by MathematicalMonk (Jeffrey Miller) for the topic.
