"the drive of doing something" is not a standard expression. 'Drive' usually suggests a stong force or effort that moves something or someone in a direction. In this context it is unlikely that the two countries are "driving" advancing world development or progress. In your sentence 'drive' most likely means "the onward course, drift, or tendency". The drive is a process that is occurring naturally. ……the two countries share extensive common interests and the basis for cooperation not only in……，but also in the "course" of advancing world development and progress. This is not a common use of the word "drive".

The "drive" is the feeling you have inside that pushes you to do something.