Does your company apply management by objectives? if yes which department is managing it?

Yes, by achieving sales targets and in the sales department

Yes of course, each department have their own OBJECTIVES, in addition to KPI(s) co-related with.

Management must be applied according to your company's objectives.
otherwise you will not succeed.
The most important department which manage this, is the department which is responsible of controlling and managing the value chain operations or core operations in your company.

I generally work as a consultant in the SME market, and from observation there is a tendency to apply MBO but not follow it.
There is a mindset to utilize people across duties to get the job done, and this does not allow the employee to grow in their specialties and have a clear-cut expectation of their value in the organization.