What kind of machine learning is this? The objective is to build a classifier that produces M correct outputs when given N inputs. 

Let a "sample" be M outputs and N inputs. Each output is some function some of the N inputs but the functions are not given. You only know that some of the N inputs can possibly combine to produce each of the M outputs.

Several samples are available for the machine learning algorithm to learn from, and in each sample, the functions for each output produced from what input sets are the exactly same.

It sounds like supervised learning but it doesn't seem to match exactly.

I'll provide an example:

__Sample 1:

Inputs: people = 6; salary = 100

outputs: width = 600; height = 100

__Sample 2:

Inputs: people = 3; salary = 60

outputs: width = 180; height = 60

__Sample 3:

Inputs: people = 5; salary = 101

outputs: width = 505; height = 101

The correct functions are then: width = people * salary; height = salary
This is supervised learning with a vector-valued output, i.e. learning a vector to vector mapping.  There's a large literature on this in the neural network community. One application is generating control signals for a physical device (e.g. pointing a camera based on visual input).  Here's a typical reference:    

Suh, I.H. and Kim, T.W. Fuzzy membership function based neural networks with applications to the visual servoing of robot manipulators.  IEEE Transactions on Fuzzy Systems, 2(3):203-220, 1994. 

From another angle, this can be viewed as multitask learning, a currently popular area in machine learning.   If one knows more about constraints among the output values, then it could usefully be viewed as a structured prediction problem, also recently popular.