When you init a project it installs cocoapods. Then when you init another project, it installs again. Why its install cocoapod again?
To keep your projects cleanly separate from one another, every project gets its own set of supporting ruby gems, as dictated by its bundler. That, indeed, is the whole point of the bundler architecture; otherwise your gems would all go into the same central ruby library, stomping on and infecting one another.
Well, cocoapods is a ruby gem. So, like every other gem for the project, it gets its own copy for this project.
(And in any case you really shouldn't worry about this. The cocoapods gem is very small.)