How to activate "Add existing field"? In the section called Manage fields within a content type, there is an option called Add an existing field. How this field is activated?

Some time ago I installed Drupal 7.38 and I worked with such option. This option was there. I never knew how it appeared there.

Now, I have two new Drupal installations (7.38 and 7.40). None of them shows by default the option called "Add an existing field".

Does the option "Add an existing field" belong to Drupal core or to some contributed module?
Add an existing field is belong to Drupal core only. 

You are not getting the option to Add an existing field because you have not created any field as of now. You cannot add an existing field until you have created it as a new field first. After you have created the new field, you can attach that field to multiple content types using Add an existing field

The Drupal.org documentation does not recommend that you re-use fields:


  "It is recommended to create new fields, rather than reusing existing
  ones, unless you have a clear reason to do so".